# Differential Structure of the Hyperbolic Clifford Algebra

Eduardo A. Notte-Cuello(1) and Waldyr A. Rodrigues Jr.(2)
(1) Departamento de Matemáticas,
Universidad de La Serena
Av. Cisternas 1200, La Serena-Chile
${}^{(2)}~{}$Institute of Mathematics, Statistics and Scientific Computation
IMECC-UNICAMP CP 6065
13083-859 Campinas, SP, Brazil
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper presents a thoughful review of: (a) the Clifford algebra
$\mathcal{C}l\left(H_{V}\right)$ of _multivecfors_ which is naturally
associated with a hyperbolic space $H_{V}$; (b) the study of the properties of
the duality product of _multivectors_ and _multiforms;_ (c) the theory of $k$
multivector and $l$ multiform variables multivector extensors over $V$ and (d)
the use of the above mentioned structures to present a theory of the
parallelism structure on an arbitrary smooth manifold introducing the concepts
of covariant derivarives, deformed covariant derivatives and relative
covariant derivatives of multivector, multiform fields and extensors fields.

Keywords: Hyperbolic Clifford algebra; multivectors, multiforms and extensor
fields; parallelism structure

##  1 Introduction

In this paper we first present a review of the Hyperbolic Clifford algebra of
a real $n$-dimensinal space $V$. We emphazise that the hyperbolic Clifford
algebra is a very important structure for modern theories of Physics, in
particular superfields can be seen as ideal sections of an hyperbolic bundel
over a spacetime $M$, see, e.g., [24, 26, 2, 13, 14, 20]. Besides given a
thoughtful exposition of the Clifford algebra $\mathcal{C}l\left(H_{V}\right)$
of _multivecfors_ which is naturally associated with a hyperbolic space
$H_{V}$, we review also the properties of the _duality product_ of
multivectors and multiforms and the theory of $k$ multivector and $l$
multiform variables multivector extensors over $V.$ The algebraic theory is
then used to review with enough details the theory of parallelism structures
on an arbitrary smooth manifold $M$, and the concepts of covariant
derivatives, deformed derivatives and relative covariant derivatives of
multivector, multiform field and extensors fields (all of crucial importance
in several physical theories, see, e.g., [12]).

##  2 Hyperbolic Spaces

###  2.1 Definition and Basic Properties

Let $V,V^{\ast}$ be a pair of dual $n$-dimensional vector spaces over the real
field $\mathbb{R}$ and $V\oplus V^{\ast}$ the exterior direct sum of the
vector spaces $V\,\ $and $V^{\ast}$. We call hyperbolic structure over $V$ to
the pair

$H_{V}=\left(V\oplus V^{\ast},\left\langle,\right\rangle\right)$

where $\left\langle,\right\rangle$ is the non-degenerate symmetric bilinear
form of index $n$ defined by

$\left\langle\mathbf{x},\mathbf{y}\right\rangle=x^{\ast}\left(y_{\ast}\right)+y^{\ast}\left(x_{\ast}\right)$
(1)

for all $\mathbf{x}=x_{\ast}\oplus x^{\ast},\mathbf{y}=y_{\ast}\oplus
y^{\ast}\in H_{V}.$

###### Definition 1

The elements of $H_{V}$ will be referred as vecfors. A vecfor
$\mathbf{x}=x_{\ast}\oplus x^{\ast}\in H_{V}$ will be said to be positive if
$x^{\ast}\left(x_{\ast}\right)>0,$ null if $x^{\ast}\left(x_{\ast}\right)=0$
and negative if $x^{\ast}\left(x_{\ast}\right)<0.$ If
$x^{\ast}\left(x_{\ast}\right)=1$ we say that $\mathbf{x}=x_{\ast}\oplus
x^{\ast}$ is a unit vecfor.

Let $H_{V}^{\ast}$ denote the dual space of $H_{V}$, i.e.,
$H_{V}^{\ast}=\left(\left(V\oplus
V^{\ast}\right)^{\ast},\left\langle,\right\rangle^{-1}\right),$ where
$\left\langle,\right\rangle^{-1}$ is the reciprocal of the bilinear form
$\left\langle,\right\rangle.$ We have the following isomorphisms

$H_{V}^{\ast}\simeq H_{V^{\ast}}\simeq H_{V}$

where $H_{V^{\ast}}$ is the hyperbolic space of $V^{\ast}.$ Therefore, $H_{V}$
is an ”auto-dual” space.

The spaces $V$ and $V^{\ast}$ are naturally identified with their images in
$H_{V}$ under the inclusions $i_{\ast}:V\rightarrow V\oplus
V^{\ast},x_{\ast}\mapsto i_{\ast}x_{\ast}=x_{\ast}\oplus 0\equiv x_{\ast}$ and
$i^{\ast}:V^{\ast}\rightarrow V\oplus V^{\ast},x^{\ast}\mapsto
i^{\ast}x^{\ast}=0\oplus x^{\ast}\equiv x^{\ast},$ then from (1) we have

$\left\langle i_{\ast}x_{\ast},i_{\ast}y_{\ast}\right\rangle=\left\langle
x_{\ast},y_{\ast}\right\rangle=0,\quad\left\langle
i^{\ast}x^{\ast},i^{\ast}y^{\ast}\right\rangle=\left\langle
x^{\ast},y^{\ast}\right\rangle=0,\quad\left\langle
i^{\ast}x^{\ast},i_{\ast}y_{\ast}\right\rangle=x^{\ast}\left(y_{\ast}\right)$

for all $x_{\ast},y_{\ast}\in V\subset V\oplus V^{\ast}$ and
$x^{\ast},y^{\ast}\in V^{\ast}\subset V\oplus V^{\ast}.$ This means that $V$
and $V^{\ast}$ are maximal totally isotropic subspace of $H_{V}$ and any pair
of dual basis satisfying

$\left\langle
e_{i},e_{j}\right\rangle=0,\quad\left\langle\theta^{i},\theta^{j}\right\rangle=0\quad\text{and}\quad\left\langle\theta^{i},e_{j}\right\rangle=\delta_{j}^{i}.$

###### Remark 2

More generally, to each subspace $S\subset V$ we can associated a maximal
totally isotropic subspace $I\left(S\right)\subset H_{V}$, see [26].

To each Witt basis
$\left\\{e_{1},\cdots,e_{n},\theta^{1},\cdots,\theta^{n}\right\\}$ of $H_{V},$
we can associated an orthonormal basis
$\left\\{\mathbf{\sigma}_{1},\cdots,\mathbf{\sigma}_{2n}\right\\}$ of $H_{V}$
by letting

$\mathbf{\sigma}_{k}=\frac{1}{\sqrt{2}}\left(e_{k}\oplus\theta^{k}\right)\quad\text{and\quad}\mathbf{\sigma}_{n+k}=\frac{1}{\sqrt{2}}\left(\overline{e}_{k}\oplus\theta^{k}\right)$
(2)

where $\overline{e}_{k}=-e_{k},k=1,...,n.$ We have for $k,l=1,...,n,$

$\left\langle\mathbf{\sigma}_{k},\mathbf{\sigma}_{l}\right\rangle=\delta_{kl},\quad\left\langle\mathbf{\sigma}_{n+k},\mathbf{\sigma}_{n+l}\right\rangle=-\delta_{kl}\quad\text{and}\quad\left\langle\mathbf{\sigma}_{k},\mathbf{\sigma}_{n+l}\right\rangle=0.$

If $\mathbf{x}=x_{\ast}\oplus x^{\ast},$ with $x_{\ast}=x_{\ast}^{k}e_{k}$ and
$x^{\ast}=x_{k}^{\ast}\theta^{k},$ then the components
$\left(x^{k},x^{n+k}\right)$ of $\mathbf{x}$ with respect to
$\mathbf{\sigma}_{k}$ are

$x^{k}=\frac{1}{\sqrt{2}}\left(x_{k}^{\ast}+x_{\ast}^{k}\right)\quad\text{and}\quad
x^{n+k}=\frac{1}{\sqrt{2}}\left(x_{k}^{\ast}-x_{\ast}^{k}\right).$ (3)

Then, base vectors $\mathbf{\sigma}_{n+l}$ are obtained from the
$\mathbf{\sigma}_{k}$ through the involution $\mathbf{x}=x_{\ast}\oplus
x^{\ast}\mapsto\overline{\mathbf{x}}=\left(-x_{\ast}\right)\oplus x^{\ast},$
where $\overline{\mathbf{x}}$ is the (hyperbolic) conjugate of the vector
$\mathbf{x.}$

Due to the auto-duality of $H_{V}$ stated by the isomorphisms
$H_{V}^{\ast}\simeq H_{V^{\ast}}\simeq H_{V}$, we can use the notation
$\left\\{\mathbf{\sigma}^{1},\cdots,\mathbf{\sigma}^{2n}\right\\}$ to indicate
the dual basis
$\left\\{\mathbf{\sigma}_{1},\cdots,\mathbf{\sigma}_{2n}\right\\}$ as well the
reciprocal basis of this same basis. As elements of $H_{V}^{\ast}$ the
expressions of the $\mathbf{\sigma}^{k\prime}$ are

$\mathbf{\sigma}^{k}=\frac{1}{\sqrt{2}}\left(\theta^{k}\oplus
e_{k}\right)\quad\text{and\quad}\mathbf{\sigma}_{k}=\frac{1}{\sqrt{2}}\left(\overline{\theta}^{k}\oplus
e_{k}\right).$

Another remarkable result on the theory of hyperbolic space is the following
(see [17]).

###### Proposition 3

Given an arbitrary non-degenerate symmetric bilinear form $b$ on $V,$ there is
an isomorphism $H_{V}\simeq\left(V,b\right)\oplus\left(V,-b\right)=H_{bV}$.

###  2.2 Exterior Algebra of a Hyperbolic Space

The Grassman algebra $\bigwedge H_{V}$ of the hyperbolic structure $H_{V}$ is
the pair

$\bigwedge H_{V}=\left(\bigwedge\left(V\oplus
V^{\ast}\right),\left\langle,\right\rangle\right)$

where

$\bigwedge\left(V\oplus
V^{\ast}\right)=\sum\limits_{r=0}^{2n}\bigwedge\nolimits^{r}\left(V\oplus
V^{\ast}\right)$

is the exterior algebra of $V\oplus V^{\ast}$ and $\left\langle,\right\rangle$
is the canonical bilinear form on $\bigwedge\left(V\oplus V^{\ast}\right)$
induced by the bilinear form $\left\langle,\right\rangle$ of $H_{V}$, e.g.,
for simple elements
$\mathbf{x}_{1}\wedge...\wedge\mathbf{x}_{r},\mathbf{y}_{1}\wedge...\wedge\mathbf{y}_{r}\in\bigwedge^{r}H_{V},\left\langle,\right\rangle$
is given by

$\left\langle\mathbf{x}_{1}\wedge...\wedge\mathbf{x}_{r},\mathbf{y}_{1}\wedge...\wedge\mathbf{y}_{r}\right\rangle=\det\left(\begin{array}[c]{ccc}\left\langle\mathbf{x}_{1},\mathbf{y}_{1}\right\rangle&\cdots&\left\langle\mathbf{x}_{1},\mathbf{y}_{r}\right\rangle\\\
.&\cdots&.\\\ .&\cdots&.\\\
\left\langle\mathbf{x}_{r},\mathbf{y}_{1}\right\rangle&\cdots&\left\langle\mathbf{x}_{r},\mathbf{y}_{r}\right\rangle\end{array}\right)$
(4)

and it is extended by linearity and orthogonality to all of the algebra
$\bigwedge H_{V}.$ Of course, due to the isomorphisms $H_{V}^{\ast}\simeq
H_{V^{\ast}}\simeq H_{V},$ we have

$\bigwedge H_{V}^{\ast}\simeq\bigwedge H_{V^{\ast}}\simeq\bigwedge H_{V},$

and it follows that

$\bigwedge H_{V}\simeq\left(\bigwedge H_{V}\right)^{\ast},$

i.e., $\bigwedge H_{V}$ is itself an auto-dual space. The elements of
$\bigwedge H_{V}$ will be called multivecfors.

Grade involution, reversion and conjugation in the algebra $\bigwedge H_{V}$
are defined as usual, see [27]. For a homogeneous multivecfors
$\mathbf{u\in}\bigwedge^{r}H_{V},$

$\widehat{\mathbf{u}}=\left(-1\right)^{r}\mathbf{u,\quad}\widetilde{\mathbf{u}}=\left(-1\right)^{\frac{1}{2}r\left(r-1\right)}\mathbf{u,\quad}\overline{\mathbf{u}}=\left(-1\right)^{\frac{1}{2}r\left(r+1\right)}\mathbf{u}$

and we call that every element $\mathbf{u\in}\bigwedge H_{V}$ is uniquely
decomposed into a sum the even and odd part of $\mathbf{u}$, i.e.,
$\mathbf{u=u}_{+}+\mathbf{u}_{-}$, where

$\mathbf{u}_{+}=\frac{1}{2}\left(\mathbf{u}+\widehat{\mathbf{u}}\right)\quad\text{and}\quad\mathbf{u}_{-}=\frac{1}{2}\left(\mathbf{u}-\widehat{\mathbf{u}}\right).$

The spaces $\bigwedge V$ and $\bigwedge V^{\ast}$ are identified with their
images in $\bigwedge H_{V}$ under the homomorphisms $i^{w}:\bigwedge
V\rightarrow\bigwedge H_{V}$ defined by

$i^{w}\left(x_{1\ast}\wedge...\wedge x_{r\ast}\right)=\left(x_{1\ast}\oplus
0\right)\wedge...\wedge\left(x_{r\ast}\oplus 0\right)\equiv
x_{1\ast}\wedge...\wedge x_{r\ast}$

and $i_{w}:\bigwedge V^{\ast}\rightarrow\bigwedge H_{V}$ defined by

$i_{w}\left(x_{1}^{\ast}\wedge...\wedge x_{r}^{\ast}\right)=\left(0\oplus
x_{1}^{\ast}\right)\wedge...\wedge\left(0\oplus x_{r}^{\ast}\right)\equiv
x_{1}^{\ast}\wedge...\wedge x_{r}^{\ast}.$

Then, for $u_{\ast},v_{\ast}\in\bigwedge V\subset\bigwedge H_{V}$ and
$u^{\ast},v^{\ast}\in\bigwedge V^{\ast}\subset\bigwedge H_{V},$ we have

$\left\langle u_{\ast},v_{\ast}\right\rangle=0,\quad\left\langle
u^{\ast},v^{\ast}\right\rangle=0\quad\text{and}\quad\left\langle
u^{\ast},v_{\ast}\right\rangle=u^{\ast}\left(v_{\ast}\right).$ (5)

Thus, $\bigwedge V$ and $\bigwedge V^{\ast}$ are totally isotropic subspace of
$\bigwedge H_{V}.$ But they are no longer maximal,
$\left\langle,\right\rangle$ being neutral, the dimension of a maximal totally
isotropic subspace is $2^{2n-1},$ whereas $\dim\bigwedge V=\dim\bigwedge
V^{\ast}=2^{n}.$ For elements $\mathbf{u}=u_{\ast}\wedge
u^{\ast},\mathbf{v}=v_{\ast}\wedge v^{\ast}\in\bigwedge H_{V}$ with
$u_{\ast}\in\bigwedge^{r}V,u^{\ast}\in\bigwedge^{s}V^{\ast},v_{\ast}\in\bigwedge^{s}V$
and $v^{\ast}\in\bigwedge^{r}V^{\ast}$ it holds

$\left\langle\mathbf{u},\mathbf{v}\right\rangle=\left(-1\right)^{rs}u^{\ast}\left(v_{\ast}\right)v^{\ast}\left(u_{\ast}\right).$

###### Proposition 4

There is the following natural isomorphism

${\textstyle\bigwedge}H_{V}\simeq{\textstyle\bigwedge}V\widehat{\otimes}{\textstyle\bigwedge}V^{\ast}$

where $\widehat{\otimes}$ denotes the graded tensor product. Moreover, being
$b$ a non-degenerate bilinear form on $V$, it holds also

${\textstyle\bigwedge}H_{V}\simeq{\textstyle\bigwedge}H_{bV}\widehat{\otimes}{\textstyle\bigwedge}H_{-bV}.$

For a proof see, e.g., Greub [15]. The first of the above isomorphisms is
given by the mapping $\bigwedge V\widehat{\otimes}\bigwedge
V^{\ast}\rightarrow\bigwedge H_{V}$ by

$u_{\ast}\widehat{\otimes}u^{\ast}\mapsto i^{w}u_{\ast}\wedge i_{w}u^{\ast},$

for all $u_{\ast}\in\bigwedge V$ and $u^{\ast}\in\bigwedge V^{\ast}$. Under
this mapping, we can make the identification

${\textstyle\bigwedge\nolimits^{r}}H_{V}={\textstyle\sum\limits_{p+q=r}}{\textstyle\bigwedge\nolimits^{p}}V\widehat{\otimes}{\textstyle\bigwedge\nolimits^{q}}V^{\ast}.$

###  2.3 Orientation

Besides a canonical metric structure, a hyperbolic space is also provided with
a canonical sense of orientation, induced by the $2n$-vector
$\mathbf{\sigma\in}\bigwedge\nolimits^{2n}H_{V}$ given by

$\mathbf{\sigma=\sigma}_{1}\wedge...\wedge\mathbf{\sigma}_{2n}$

where $\left\\{\mathbf{\sigma}_{1},\cdots,\mathbf{\sigma}_{2n}\right\\}$ is
the orthonormal basis of $H_{V}$ naturally associated with the dual basis
$\left\\{e_{1},\cdots,e_{n}\right\\}$ of $V$ and
$\left\\{\theta^{1},\cdots,\theta^{n}\right\\}$ of $V^{\ast}.$ Note that from
Eq.(4) it follows immediately that

$\left\langle\mathbf{\sigma,\sigma}\right\rangle=\left(-1\right)^{n}.$

The reason for saying that $\mathbf{\sigma}$ define a canonical sense of
orientation is that it is independent on the choice of the bases for $V.$ To
see this it is enough to verify that

$\mathbf{\sigma}=e_{\ast}\wedge\theta^{\ast}$

where

$e_{\ast}=e_{1}\wedge\cdots\wedge
e_{n}\quad\text{and\quad}\theta^{\ast}=\theta^{1}\wedge\cdots\wedge\theta^{n}.$

Thus, under a change of basis in $V,$ $e_{\ast}$ transforms as $\lambda
e_{\ast},$ with $\lambda\neq 0,$ whereas $\theta^{\ast}$ transforms as
$\lambda^{-1}\theta^{\ast},$ so that $\mathbf{\sigma}$ remains unchanged.

###  2.4 Contractions

A left contraction
$\lrcorner:{\textstyle\bigwedge}H_{V}\times{\textstyle\bigwedge}H_{V}\rightarrow{\textstyle\bigwedge}H_{V}$
and a right contraction
$\llcorner:{\textstyle\bigwedge}H_{V}\times{\textstyle\bigwedge}H_{V}\rightarrow{\textstyle\bigwedge}H_{V}$
are introduced in the algebra ${\textstyle\bigwedge}H_{V}$ in the usual way
(see, e.g., [27]), i.e., by

$\left\langle\mathbf{u}\lrcorner\mathbf{v},\mathbf{w}\right\rangle=\left\langle\mathbf{v},\widetilde{\mathbf{u}}\wedge\mathbf{w}\right\rangle\quad
and\quad\left\langle\mathbf{v}\llcorner\mathbf{u},\mathbf{w}\right\rangle=\left\langle\mathbf{v},\mathbf{w}\wedge\widetilde{\mathbf{u}}\right\rangle,$

for all $\mathbf{u,v,w\in}{\textstyle\bigwedge}H_{V}.$ These operations have
the general properties, for all $\mathbf{x,y\in}H_{V}$, for all
$\mathbf{u,v,w\in}{\textstyle\bigwedge}H_{V}$ and
$\mathbf{\sigma\in}\bigwedge\nolimits^{2n}H_{V},$

$\begin{array}[c]{l}\mathbf{x}\lrcorner\mathbf{y}=\mathbf{x}\llcorner\mathbf{y=}\left\langle\mathbf{x},\mathbf{y}\right\rangle\quad\text{;}\quad
1\lrcorner\mathbf{u=u}\llcorner=\mathbf{u},\quad\mathbf{x}\lrcorner=1\llcorner\mathbf{x}=0\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\left(\mathbf{u}\lrcorner\mathbf{v}\right)^{\wedge}=\widehat{\mathbf{u}}\lrcorner\widehat{\mathbf{v}},\quad\left(\mathbf{u}\llcorner\mathbf{v}\right)^{\wedge}=\widehat{\mathbf{u}}\llcorner\widehat{\mathbf{v}}\quad\text{;}\quad\left(\mathbf{u}\lrcorner\mathbf{v}\right)^{\sim}=\widetilde{\mathbf{u}}\lrcorner\widetilde{\mathbf{v}},\quad\left(\mathbf{u}\llcorner\mathbf{v}\right)^{\sim}=\widetilde{\mathbf{u}}\llcorner\widetilde{\mathbf{v}}\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\mathbf{u}\lrcorner\left(\mathbf{v}\lrcorner\mathbf{w}\right)=\left(\mathbf{u}\wedge\mathbf{v}\right)\lrcorner\mathbf{w;}\text{
}\left(\mathbf{u}\llcorner\mathbf{v}\right)\llcorner\mathbf{w}=\mathbf{u}\llcorner\left(\mathbf{v}\wedge\mathbf{w}\right)\mathbf{,}\text{
}\left(\mathbf{u}\lrcorner\mathbf{v}\right)\llcorner\mathbf{w}=\mathbf{u}\lrcorner\left(\mathbf{v}\llcorner\mathbf{w}\right)\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\mathbf{x}\lrcorner\left(\mathbf{v}\wedge\mathbf{w}\right)=\left(\mathbf{x}\lrcorner\mathbf{v}\right)\wedge\mathbf{w+}\widehat{\mathbf{v}}\wedge\left(\mathbf{x\lrcorner
w}\right)\quad\text{;}\quad\left(\mathbf{u}\wedge\mathbf{v}\right)\llcorner\mathbf{x}=\mathbf{u\wedge}\left(\mathbf{\mathbf{v}}\llcorner\mathbf{x}\right)\mathbf{+}\left(\mathbf{u\llcorner
x}\right)\wedge\widehat{\mathbf{v}}\vskip 12.0pt plus 4.0pt minus 4.0pt\\\
\mathbf{x}\wedge\left(\mathbf{v}\lrcorner\mathbf{w}\right)=\widehat{\mathbf{v}}\lrcorner\left(\mathbf{x\wedge
w}\right)-\left(\widehat{\mathbf{v}}\llcorner\mathbf{x}\right)\lrcorner\mathbf{w}\quad\text{;}\quad\left(\mathbf{u\llcorner
v}\right)\wedge\mathbf{x}=\left(\mathbf{u\wedge
x}\right)\llcorner\widehat{\mathbf{v}}\mathbf{-u\llcorner}\left(\mathbf{x}\lrcorner\widehat{\mathbf{v}}\right)\vskip
12.0pt plus 4.0pt minus 4.0pt\\\ \mathbf{u}_{+}\lrcorner\mathbf{v=v\llcorner
u}_{+}\quad\text{;}\quad\mathbf{u}_{-}\lrcorner\mathbf{v=}\widehat{\mathbf{v}}\mathbf{\llcorner}\widehat{\mathbf{u}}_{-}\vskip
12.0pt plus 4.0pt minus 4.0pt\\\
\mathbf{u}\wedge\left(\mathbf{v}\lrcorner\mathbf{\sigma}\right)=\left(\mathbf{u}\lrcorner\mathbf{v}\right)\lrcorner\mathbf{\sigma}\quad\text{;}\quad\left(\mathbf{\sigma\llcorner
v}\right)\wedge\mathbf{w=\sigma\llcorner}\left(\mathbf{v}\llcorner\mathbf{w}\right)\end{array}$

Moreover, from Eqs.(5) we get

$u_{\ast}\lrcorner v_{\ast}=u_{\ast}\llcorner v_{\ast}=0,\quad
u^{\ast}\lrcorner v^{\ast}=u^{\ast}\llcorner v^{\ast}=0$ (6)

for all $u_{\ast},v_{\ast}\mathbf{\in}{\textstyle\bigwedge}V$ and
$u^{\ast},v^{\ast}\mathbf{\in}{\textstyle\bigwedge}V^{\ast},$ so that for
elements of the form $\mathbf{u}=u_{\ast}\wedge u^{\ast}$ and
$\mathbf{x}=x_{\ast}\oplus x^{\ast},$ it holds

$\begin{array}[c]{c}\mathbf{x}\lrcorner\mathbf{u}=\left(x^{\ast}\lrcorner
u_{\ast}\right)\wedge u^{\ast}+\widehat{u}_{\ast}\wedge\left(x_{\ast}\lrcorner
u^{\ast}\right)\vskip 6.0pt plus 2.0pt minus 2.0pt\\\
\mathbf{u}\llcorner\mathbf{x}=u_{\ast}\wedge\left(u^{\ast}\llcorner
x_{\ast}\right)+\left(x^{\ast}\lrcorner
u_{\ast}\right)\wedge\widehat{u}^{\ast}\end{array}$

For more details about the properties of the left and right contractions, see,
e.g., [19, 27].

###  2.5 Poincaré Automorphism (Hodge Dual)

Define now the Poincaré automorphism or Hodge dual
$\star:{\textstyle\bigwedge}H_{V}\rightarrow{\textstyle\bigwedge}H_{V}$ by

$\star\mathbf{u=}\widetilde{\mathbf{u}}\lrcorner\mathbf{\sigma,}$ (7)

for all $\mathbf{u\in}{\textstyle\bigwedge}H_{V}.$ The inverse $\star^{-1}$ of
this operation is given by

$\star^{-1}\mathbf{u=}\widetilde{\mathbf{\sigma}}\mathbf{\llcorner\widetilde{\mathbf{u}}.}$
(8)

The following general properties of the Hodge duality holds true

$\begin{array}[c]{l}\star\mathbf{\sigma=}\left(-1\right)^{n},\quad\star^{-1}\mathbf{\sigma=}1,\quad\left\langle\star\mathbf{u,}\star\mathbf{v}\right\rangle=\left(-1\right)^{n}\left\langle\mathbf{u,v}\right\rangle,\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \star\left(\mathbf{u\wedge
v}\right)=\widetilde{\mathbf{v}}\lrcorner\star\mathbf{u},\quad\star^{-1}\left(\mathbf{u\wedge
v}\right)=\left(\star^{-1}\mathbf{v}\right)\llcorner\widetilde{\mathbf{u,}}\vskip
6.0pt plus 2.0pt minus 2.0pt\\\ \star\left(\mathbf{u\llcorner
v}\right)=\widetilde{\mathbf{v}}\wedge\star\mathbf{u},\quad\star^{-1}\left(\mathbf{u\lrcorner
v}\right)=\left(\star^{-1}\mathbf{v}\right)\wedge\widetilde{\mathbf{u}}.\end{array}$

For $\mathbf{x}=x_{\ast}\oplus x^{\ast}\in
H_{V}\subset{\textstyle\bigwedge}H_{V}$ we have, since $x_{\ast}\lrcorner
e_{\ast}=0$ and $x^{\ast}\lrcorner\theta^{\ast}=0$ that

$\begin{array}[c]{l}\star\mathbf{x=x\lrcorner\sigma=x\lrcorner}\left(e_{\ast}\wedge\theta^{\ast}\right)=\left(\mathbf{x\lrcorner}e_{\ast}\right)\wedge\theta^{\ast}+\widehat{e}_{\ast}\wedge\left(\mathbf{x\lrcorner}\theta^{\ast}\right)\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
=\left(x^{\ast}\mathbf{\lrcorner}e_{\ast}\right)\wedge\theta^{\ast}-e_{\ast}\wedge\left(\theta^{\ast}\llcorner
x_{\ast}\right),\end{array}$

and it follows that, for
$u_{\ast}\in{\textstyle\bigwedge}V\subset{\textstyle\bigwedge}H_{V}$ and
$u^{\ast}\in{\textstyle\bigwedge}V^{\ast}\subset{\textstyle\bigwedge}H_{V},$

$\star
u^{\ast}\mathbf{=}\left(\widetilde{u}^{\ast}\mathbf{\lrcorner}e_{\ast}\right)\wedge\theta^{\ast}=D_{\nshortparallel}u^{\ast}\wedge\theta^{\ast}\quad\text{and}\quad\star
u_{\ast}\mathbf{=}e_{\ast}\wedge\left(\theta^{\ast}\llcorner\overline{u}_{\ast}\right)=e_{\ast}\wedge
D^{\nshortparallel}u_{\ast}$

where we introduced the Poincaré isomorphisms
$D_{\nshortparallel}:{\textstyle\bigwedge}V^{\ast}\rightarrow{\textstyle\bigwedge}V$
and
$D^{\nshortparallel}:{\textstyle\bigwedge}V\rightarrow{\textstyle\bigwedge}V^{\ast}$
by (see, e.g., [15])

$D_{\nshortparallel}u^{\ast}=\widetilde{u}^{\ast}\mathbf{\lrcorner}e_{\ast}\quad\text{and}\quad
D^{\nshortparallel}u_{\ast}=\theta^{\ast}\llcorner\overline{u}_{\ast}.$

For an element of the form $\mathbf{u}=u_{\ast}\wedge u^{\ast}$ with
$u_{\ast}\in{\textstyle\bigwedge}V$ and
$u^{\ast}\in{\textstyle\bigwedge}V^{\ast},$ we have

$\star\mathbf{u=}D_{\nshortparallel}u^{\ast}\wedge
D^{\nshortparallel}u_{\ast}.$

###  2.6 Clifford Algebra of a Hyperbolic Space

Introduce in ${\displaystyle\bigwedge}H_{V}$ the Clifford product of a vector
$\mathbf{x}\in H_{V}$ by an element $\mathbf{u}\in{\textstyle\bigwedge}H_{V}$
by

$\mathbf{xu}=\mathbf{x}\lrcorner\mathbf{u}+\mathbf{x}\wedge\mathbf{u}$

and extend this product by linearity and associativity to all of the space
${\displaystyle\bigwedge}H_{V}$. The resulting algebra is isomorphic to the
Clifford algebra $\mathcal{C\ell}(H_{V})$ of the hyperbolic structure $H_{V}$
and will thereby be identified with it.

We call $\mathcal{C\ell}(H_{V})$ the _mother_ algebra(or the hyperbolic
Clifford algebra) of the vector space $V$. The even and odd subspaces of
$\mathcal{C\ell}(H_{V})$ will be denoted respectively by
$\mathcal{C\ell}^{(0)}(H_{V})$ and $\mathcal{C\ell}^{(1)}(H_{V})$, so that

$\mathcal{C\ell}(H_{V})=\mathcal{C\ell}^{(0)}(H_{V})\oplus\mathcal{C\ell}^{(1)}(H_{V})$

and the same notation of the exterior algebra is used for grade involution,
reversion, and conjugation in $\mathcal{C\ell}(H_{V})$, which obviously
satisfy

$(\mathbf{uv})\hat{\
}=\hat{\mathbf{u}}\hat{\mathbf{v}},\qquad(\mathbf{uv})\tilde{\
}=\tilde{\mathbf{v}}\tilde{\mathbf{u}},\qquad(\mathbf{uv})\bar{\
}=\bar{\mathbf{v}}\bar{\mathbf{u}}.$

For vecfors $\mathbf{x},\mathbf{y}\in H_{V}$, we have the relation

$\mathbf{xy}+\mathbf{yx}=2\boldsymbol{<}\mathbf{x},\mathbf{y}>.$

For the basis elements $\\{\mathbf{\sigma}_{k}\\}$ it holds

$\begin{array}[c]{l}\mathbf{\sigma}_{k}\mathbf{\sigma}_{l}+\mathbf{\sigma}_{l}\mathbf{\sigma}_{k}=2\delta_{kl},\\\
\mathbf{\sigma}_{n+k}\mathbf{\sigma}_{n+l}+\mathbf{\sigma}_{n+l}\mathbf{\sigma}_{n+k}=-2\delta_{kl},\\\
\mathbf{\sigma}_{k}\mathbf{\sigma}_{n+l}=-\mathbf{\sigma}_{n+l}\mathbf{\sigma}_{k}.\end{array}$

In turn, for the Witt basis $\\{e_{k},\theta^{k}\\}$, we have instead

$\displaystyle e_{k}e_{l}+e_{l}e_{k}=0,$
$\displaystyle\theta^{k}\theta^{l}+\theta^{l}\theta^{k}=0,$
$\displaystyle\theta^{k}e_{l}+e_{l}\theta^{k}=2\delta_{l}^{k}.$

The Clifford product has the following general properties:

$\begin{array}[c]{l}\mathbf{u\lrcorner\sigma=u\sigma,\qquad\sigma\llcorner
u=\sigma u,\vskip 6.0pt plus 2.0pt minus 2.0pt}\\\
\left\langle\mathbf{u,vw}\right\rangle=\left\langle\widetilde{\mathbf{v}}\mathbf{u,w}\right\rangle=\left\langle\mathbf{u}\widetilde{\mathbf{w}}\mathbf{,v}\right\rangle,\mathbf{\vskip
6.0pt plus 2.0pt minus 2.0pt}\\\ \mathbf{x\wedge
u=}\frac{1}{2}\left(\mathbf{xu+}\widehat{\mathbf{u}}\mathbf{x}\right)\mathbf{,\qquad
u\wedge
x=}\frac{1}{2}\left(\mathbf{ux+x}\widehat{\mathbf{u}}\right),\mathbf{\vskip
6.0pt plus 2.0pt minus 2.0pt}\\\ \mathbf{x\lrcorner
u=}\frac{1}{2}\left(\mathbf{xu-}\widehat{\mathbf{u}}\mathbf{x}\right)\mathbf{,\qquad
u\llcorner
x=}\frac{1}{2}\left(\mathbf{ux-x}\widehat{\mathbf{u}}\right),\mathbf{\vskip
6.0pt plus 2.0pt minus 2.0pt}\\\
\mathbf{x\lrcorner}\left(\mathbf{uv}\right)\mathbf{=}\left(\mathbf{x\lrcorner
u}\right)\mathbf{v+}\widehat{\mathbf{u}}\left(\mathbf{x\lrcorner
v}\right)\mathbf{,\qquad}\left(\mathbf{uv}\right)\mathbf{\llcorner
x=u}\left(\mathbf{v\llcorner x}\right)\mathbf{-}\left(\mathbf{u\llcorner
x}\right)\widehat{\mathbf{v}},\mathbf{\vskip 6.0pt plus 2.0pt minus 2.0pt}\\\
\mathbf{x\wedge}\left(\mathbf{uv}\right)\mathbf{=}\left(\mathbf{x\lrcorner
u}\right)\mathbf{v+}\widehat{\mathbf{u}}\left(\mathbf{x\wedge
v}\right)=\left(\mathbf{x\wedge
u}\right)\mathbf{v-}\widehat{\mathbf{u}}\left(\mathbf{x\lrcorner
v}\right),\mathbf{\vskip 6.0pt plus 2.0pt minus 2.0pt}\\\
\left(\mathbf{uv}\right)\wedge\mathbf{x=u}\left(\mathbf{v\wedge
x}\right)-\left(\mathbf{u\llcorner
x}\right)\widehat{\mathbf{v}}=\mathbf{u}\left(\mathbf{v\llcorner
x}\right)\mathbf{+}\left(\mathbf{u\wedge
x}\right)\widehat{\mathbf{v}},\mathbf{\vskip 6.0pt plus 2.0pt minus 2.0pt}\\\
\star\mathbf{u}=\widetilde{\mathbf{u}}\mathbf{\sigma,\qquad}\star^{-1}\mathbf{u}=\widetilde{\mathbf{\sigma}}\widetilde{\mathbf{u}},\mathbf{\vskip
6.0pt plus 2.0pt minus 2.0pt}\\\
\star\left(\mathbf{uv}\right)=\widetilde{\mathbf{v}}\left(\star\mathbf{u}\right)\mathbf{,\qquad}\star^{-1}\left(\mathbf{uv}\right)=\left(\star^{-1}\mathbf{v}\right)\mathbf{u.}\\\
\end{array}$

Moreover, for an element of the form $\mathbf{u}=u_{\ast}\wedge u^{\ast}$,
with $u_{\ast}\in{\textstyle\bigwedge}V$ and
$u^{\ast}\in{\textstyle\bigwedge}V^{\ast}$ and
$\mathbf{x\in}{\textstyle\bigwedge}H_{V}$ it is:

$\displaystyle\mathbf{xu}$ $\displaystyle\mathbf{=}\left(x_{\ast}\oplus
x^{\ast}\right)\lrcorner\left(u_{\ast}\wedge
u^{\ast}\right)+\left(x_{\ast}\oplus x^{\ast}\right)\wedge\left(u_{\ast}\wedge
u^{\ast}\right)\vskip 6.0pt plus 2.0pt minus 2.0pt$
$\displaystyle=\left(x^{\ast}\lrcorner u_{\ast}\right)\wedge
u^{\ast}+\widehat{u}_{\ast}\wedge\left(x_{\ast}\lrcorner
u^{\ast}\right)+\left(x_{\ast}\oplus x^{\ast}\right)\wedge\left(u_{\ast}\wedge
u^{\ast}\right).$

On the other hand,

$\left(x_{\ast}\oplus x^{\ast}\right)\wedge\left(u_{\ast}\wedge
u^{\ast}\right)=\widehat{u}_{\ast}\wedge\left(x_{\ast}\wedge
u^{\ast}\right)+\left(x^{\ast}\wedge u_{\ast}\right)\wedge u^{\ast}$

and then,

$\displaystyle\mathbf{xu}$ $\displaystyle\mathbf{=}\left(x^{\ast}\lrcorner
u_{\ast}\right)\wedge u^{\ast}+\widehat{u}_{\ast}\wedge\left(x_{\ast}\lrcorner
u^{\ast}\right)+\widehat{u}_{\ast}\wedge\left(x_{\ast}\wedge
u^{\ast}\right)+\left(x^{\ast}\wedge u_{\ast}\right)\wedge u^{\ast}\vskip
6.0pt plus 2.0pt minus 2.0pt$ $\displaystyle=\left(\left(x^{\ast}\lrcorner
u_{\ast}\right)+\left(x^{\ast}\wedge u_{\ast}\right)\right)\wedge
u^{\ast}+\widehat{u}_{\ast}\wedge\left(\left(x_{\ast}\lrcorner
u^{\ast}\right)+\left(x_{\ast}\wedge u^{\ast}\right)\right)\vskip 6.0pt plus
2.0pt minus 2.0pt$ $\displaystyle=(x^{\ast}u_{\ast})\wedge
u^{\ast}+\hat{u}_{\ast}\wedge(x_{\ast}u^{\ast}).$

Also note that the square of the volume $2n$-vector $\mathbf{\sigma}$ satisfy

$\mathbf{\sigma}^{2}=1.$

###### Proposition 5

There is the following natural isomorphism

$\mathcal{C\ell}(H_{V})\simeq\mathop{\mathrm{End}}({\displaystyle\bigwedge}V).$

In addition, being $b$ a non-degenerate symmetric bilinear form on $V$, it
holds also

$\mathcal{C\ell}(H_{V})\simeq\mathcal{C\ell}(H_{bV})\simeq\mathcal{C\ell}(V,b)\text{
}\widehat{\otimes}\text{ }\mathcal{C\ell}(V,-b).$

Proof. The first isomorphism is given by the extension to
$\mathcal{C\ell}(H_{V})$ of the Clifford map
$\varphi:H_{V}\rightarrow\mathop{\mathrm{End}}({\displaystyle\bigwedge}V$ $)$
by $\mathbf{x}\mapsto\varphi_{x}$, with

$\varphi_{x}(u_{\ast})={\frac{1}{\sqrt{2}}}\left(x^{\ast}\mathbin{\lrcorner}u_{\ast}+x_{\ast}\wedge
u_{\ast}\right),$

for all $u_{\ast}\in{\displaystyle\bigwedge}V$. The second isomorphism, in
turn, is induced from the Clifford map

$x_{\ast}\oplus x^{\ast}\mapsto
x_{+}\widehat{\otimes}1+1\widehat{\otimes}x_{-},$

with

$x_{\pm}={{\frac{1}{\sqrt{2}}}}(b^{\ast}x^{\ast}\pm x_{\ast}).$

###### Corollary 6

The even and odd subspaces of the hyperbolic Clifford algebra are

$\mathcal{C\ell}^{(0)}(H_{V})\simeq\mathop{\mathrm{End}}({\displaystyle\bigwedge\nolimits^{(0)}}V)\oplus\mathop{\mathrm{End}}({\displaystyle\bigwedge\nolimits^{(1)}}V)$

and

$\mathcal{C\ell}^{(1)}(H_{V})\simeq
L({\displaystyle\bigwedge\nolimits^{(0)}}V,{\displaystyle\bigwedge\nolimits^{(1)}}V)\oplus
L({\displaystyle\bigwedge\nolimits^{(1)}}{\displaystyle\bigwedge},{\displaystyle\bigwedge\nolimits^{(0)}}V)$

where $L(V,W)$ denotes the space of the linear mappings from $V$ to $W$ and
${\textstyle\bigwedge\nolimits^{\left(0\right)}}V$ and
${\textstyle\bigwedge\nolimits^{\left(1\right)}}V$ denote respectively the
spaces of the even and of odd elements of ${\textstyle\bigwedge}V.$

###### Remark 7

Recalling the definition of the second order hyperbolic structure,
$H_{V}^{2}$, it follows from propositon 5 that

$\mathcal{C\ell}(H_{V}^{2})\simeq\mathop{\mathrm{End}}(\mathcal{C\ell}(H_{V})$

The algebra $\mathcal{C\ell}(H_{V}^{2})$ may be called grandmother algebraof
the vector space $V$.

##  3 Duality Products of Multivectors and Multiforms, and Extensors

In this section we study the _duality product_ of multivectors by multiforms
used in the definition of the hyperbolic algebra $\mathcal{C\ell}(V\oplus
V^{\ast},\left\langle,\right\rangle)$ of _multivecfors_. We detail some
important properties of the left and right contracted products among the
elements of ${\textstyle\bigwedge}V$ and ${\textstyle\bigwedge}V^{\ast},$by
introducing a very useful notation for these products. Next, we give a theory
of the $k$_multivector and_ $l$_multiform variables multivector_(_or
multiform_)_extensors_ over $V$ (defining the spaces
$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)$ and
$\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)$) introducing the
concept of exterior product of extensors, and of several operators acting on
these objects as, e.g., the _adjoint operator_ , the _exterior power extension
operator_ and the _contracted extension operator_. We analyze the properties
of these operators with considerable detail.

###  3.1 Duality Scalar Product of Multivectors and Multiforms

In the Appendix A we briefly recall the exterior algebras of multivectors
(elements of ${\textstyle\bigwedge}V$) and multiforms (elements of
${\textstyle\bigwedge}V^{\ast}$) associated with a real vector space $V$ of
finite dimension which is need for the following.

###### Definition 8

The duality scalar product of a multiform $\Phi$ with a multivector $X$ is the
scalar $\left\langle\Phi,X\right\rangle$ defined by the following axioms:

For all $\alpha,\beta\in\mathbb{R}:$

$\left\langle\alpha,\beta\right\rangle=\left\langle\beta,\alpha\right\rangle=\alpha\beta.$
(9)

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ (with $1\leq p\leq n$)$:$

$\left\langle\Phi_{p},X^{p}\right\rangle=\left\langle
X^{p},\Phi_{p}\right\rangle=\frac{1}{p!}\Phi_{p}(e_{j_{1}},\ldots,e_{j_{p}})X^{p}(\varepsilon^{j_{1}},\ldots,\varepsilon^{j_{p}}),$
(10)

where $\left\\{e_{j},\varepsilon^{j}\right\\}$ is any pair of dual bases over
$V.$

For all $\Phi\in{\textstyle\bigwedge}V^{\ast}$ and
$X\in{\textstyle\bigwedge}V:$ if $\Phi=\Phi_{0}+\Phi_{1}+\cdots+\Phi_{n}$ and
$X=X^{0}+X^{1}+\cdots+X^{n},$ then

$\left\langle\Phi,X\right\rangle=\left\langle
X,\Phi\right\rangle=\overset{n}{\underset{p=0}{{\textstyle\sum}}}\left\langle\Phi_{p},X^{p}\right\rangle.$
(11)

We emphasize that the scalar
$\Phi_{p}(e_{j_{1}},\ldots,e_{j_{p}})X^{p}(\varepsilon^{j_{1}},\ldots,\varepsilon^{j_{p}})$
has frame independent character, i.e., it does not depend on the pair of dual
bases $\left\\{e_{j},\varepsilon^{j}\right\\}$ used in its evaluation, since
$\Phi_{p}$ and $X^{p}$ are $p$-linear mappings.

Note that for all $\omega\in V^{\ast}$ and $v\in V$ it holds

$\left\langle
v,\omega\right\rangle=\left\langle\omega,v\right\rangle=\omega(v).$ (12)

We present now two noticeable properties for the duality scalar product
between $p$-forms and $p$-vectors:

(i) For all $\Phi_{p}\in{\textstyle\bigwedge^{p}}V^{\ast},$ and
$v_{1},\ldots,v_{p}\in V:$

$\left\langle\Phi_{p},v_{1}\wedge\cdots\wedge v_{p}\right\rangle=\left\langle
v_{1}\wedge\cdots\wedge
v_{p},\Phi_{p}\right\rangle=\Phi_{p}(v_{1},\ldots,v_{p}).$ (13)

(ii) For all $\omega^{1},\ldots,\omega^{p}\in V^{\ast}$ and
$v_{1},\ldots,v_{p}\in V:$

$\left\langle\omega^{1}\wedge\cdots\wedge\omega^{p},v_{1}\wedge\cdots\wedge
v_{p}\right\rangle=\det\left(\begin{array}[c]{lll}\omega^{1}(v_{1})&\cdots&\omega^{1}(v_{p})\\\
\vdots&\vdots&\vdots\\\
\omega^{p}(v_{1})&\cdots&\omega^{p}(v_{p})\end{array}\right).$ (14)

The basic properties for the duality scalar product are the non-degeneracy and
the distributive laws on the left and on the right with respect to addition of
either multiforms or multivectors, i.e.,

$\displaystyle\left\langle\Phi,X\right\rangle$ $\displaystyle=0,\text{ for all
}\Phi,\text{ then }X=0,$ $\displaystyle\left\langle\Phi,X\right\rangle$
$\displaystyle=0,\text{ for all }X,\text{ then }\Phi=0,$ (15)

$\displaystyle\left\langle\Phi+\Psi,X\right\rangle$
$\displaystyle=\left\langle\Phi,X\right\rangle+\left\langle\Psi,X\right\rangle,$
$\displaystyle\left\langle\Phi,X+Y\right\rangle$
$\displaystyle=\left\langle\Phi,X\right\rangle+\left\langle\Phi,Y\right\rangle.$
(16)

###  3.2 Duality Contracted Products of Multivectors and Multiforms

####  3.2.1 Left Contracted Product

###### Definition 9

The left contracted product of a multiform $\Phi$ with a multivector $X$ (or,
a multivector $X$ with a multiform $\Phi$) is the multivector
$\left\langle\Phi,X\right|$ (respectively, the multiform $\left\langle
X,\Phi\right|$) defined by the following axioms:

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ with $0\leq p\leq n:$

$\left\langle\Phi_{p},X^{p}\right|=\left\langle
X^{p},\Phi_{p}\right|=\left\langle\widetilde{\Phi}_{p},X^{p}\right\rangle=\left\langle\Phi_{p},\widetilde{X}^{p}\right\rangle.$
(17)

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{q}\in{\textstyle\bigwedge\nolimits^{q}}V$ (or
$X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ and
$\Phi_{q}\in{\textstyle\bigwedge\nolimits^{q}}V^{\ast}$) with $0\leq p<q\leq
n:$

$\displaystyle\left\langle\Phi_{p},X^{q}\right|$
$\displaystyle=\frac{1}{(q-p)!}\left\langle\widetilde{\Phi}_{p}\wedge\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}},X^{q}\right\rangle
e_{j_{1}}\wedge\cdots\wedge e_{j_{q-p}},$ (18) $\displaystyle\left\langle
X^{p},\Phi_{q}\right|$
$\displaystyle=\frac{1}{(q-p)!}\left\langle\widetilde{X}^{p}\wedge
e_{j_{1}}\wedge\cdots\wedge
e_{j_{q-p}},\Phi_{q}\right\rangle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}},$
(19)

where $\left\\{e_{j},\varepsilon^{j}\right\\}$ is any pair of dual bases for
$V$ and $V^{\ast}$.

For all $\Phi\in{\textstyle\bigwedge}V^{\ast}$ and
$X\in{\textstyle\bigwedge}V:$ if $\Phi=\Phi_{0}+\Phi_{1}+\cdots+\Phi_{n}$ and
$X=X^{0}+X^{1}+\cdots+X^{n},$ then

$\displaystyle\left\langle\Phi,X\right|$
$\displaystyle=\overset{n}{\underset{k=0}{{\textstyle\sum}}}\underset{j=0}{\overset{n-k}{{\textstyle\sum}}}\left\langle\Phi_{j},X^{k+j}\right|,$
(20) $\displaystyle\left\langle X,\Phi\right|$
$\displaystyle=\overset{n}{\underset{k=0}{{\textstyle\sum}}}\underset{j=0}{\overset{n-k}{{\textstyle\sum}}}\left\langle
X^{j},\Phi_{k+j}\right|.$ (21)

Note that the $(q-p)$-vector $\left\langle\Phi_{p},X^{q}\right|$ and the
$(q-p)$-form $\left\langle X^{p},\Phi_{q}\right|$ have frame independent
character, i.e., they do not depend on the pair of frames
$\left\\{e_{j},\varepsilon^{j}\right\\}$ chosen for calculating them.

The left contracted product has the following basic properties:

###### Proposition 10

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{q}\in{\textstyle\bigwedge\nolimits^{q}}V$ with $0\leq p\leq q\leq n.$ For
all $\Psi_{q-p}\in{\textstyle\bigwedge\nolimits^{q-p}}V^{\ast}$, it holds

$\left\langle\left\langle\Phi_{p},X^{q}\right|,\Psi_{q-p}\right\rangle=\left\langle
X^{q},\widetilde{\Phi}_{p}\wedge\Psi_{q-p}\right\rangle.$ (22)

For all $X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ and
$\Phi_{q}\in{\textstyle\bigwedge\nolimits^{q}}V^{\ast}$ with $0\leq p\leq
q\leq n.$ For all $Y^{q-p}\in{\textstyle\bigwedge\nolimits^{q-p}}V$, it holds

$\left\langle\left\langle
X^{p},\Phi_{q}\right|,Y^{q-p}\right\rangle=\left\langle\Phi_{q},\widetilde{X}^{p}\wedge
Y^{q-p}\right\rangle.$ (23)

For all $X\in{\textstyle\bigwedge}V$ and
$\Phi,\Psi\in{\textstyle\bigwedge}V^{\ast}:$

$\left\langle\left\langle\Phi,X\right|,\Psi\right\rangle=\left\langle
X,\widetilde{\Phi}\wedge\Psi\right\rangle.$ (24)

For all $\Phi\in{\textstyle\bigwedge}V^{\ast}$ and
$X,Y\in{\textstyle\bigwedge}V:$

$\left\langle\left\langle
X,\Phi\right|,Y\right\rangle=\left\langle\Phi,\widetilde{X}\wedge
Y\right\rangle.$ (25)

###### Proposition 11

The left contracted product satisfies the distributive laws on the left and on
the right.

For all $\Phi,\Psi\in{\textstyle\bigwedge}V^{\ast}$ and
$X,Y\in{\textstyle\bigwedge}V:$

$\displaystyle\left\langle(\Phi+\Psi),X\right|$
$\displaystyle=\left\langle\Phi,X\right|+\left\langle\Psi,X\right|,$
$\displaystyle\left\langle\Phi,(X+Y)\right|$
$\displaystyle=\left\langle\Phi,X\right|+\left\langle\Phi,Y\right|.$ (26)

For all $X,Y\in{\textstyle\bigwedge}V$ and
$\Phi,\Psi\in{\textstyle\bigwedge}V^{\ast}:$

$\displaystyle\left\langle(X+Y),\Phi\right|$ $\displaystyle=\left\langle
X,\Phi\right|+\left\langle Y,\Phi\right|,$ $\displaystyle\left\langle
X,(\Phi+\Psi)\right|$ $\displaystyle=\left\langle X,\Phi\right|+\left\langle
X,\Psi\right|.$ (27)

Proof. We present only the proof of the property given by Eq.(22), the other
proofs being somewhat analogous.

First note that if $\Psi_{q-p}\in{\textstyle\bigwedge\nolimits^{q-p}}V^{\ast}$
and $\left\\{e_{j},\varepsilon^{j}\right\\}$ is any pair of dual bases for $V$
and $V^{\ast},$ we can write

$\Psi_{q-p}=\frac{1}{\left(q-p\right)!}\left\langle\Psi_{q-p},e_{j_{1}}\wedge...\wedge
e_{j_{q-p}}\right\rangle\varepsilon^{j_{1}}\wedge...\wedge\varepsilon^{j_{q-p}}.$

Then, using the axiom of the Eq.(18) and the above equation we have

$\begin{array}[c]{l}\left\langle\left\langle\Phi_{p},X^{q}\right|,\Psi_{q-p}\right\rangle\\\
=\frac{1}{\left(q-p\right)!}\left\langle\left\langle\widetilde{\Phi}_{p}\wedge\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}},X^{q}\right\rangle
e_{j_{1}}\wedge\cdots\wedge e_{j_{q-p}},\Psi_{q-p}\right\rangle\\\
=\frac{1}{\left(q-p\right)!}\left\langle\widetilde{\Phi}_{p}\wedge\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}},X^{q}\right\rangle\left\langle
e_{j_{1}}\wedge\cdots\wedge e_{j_{q-p}},\Psi_{q-p}\right\rangle\\\
=\left\langle
X^{q},\frac{1}{\left(q-p\right)!}\left\langle\Psi_{q-p},e_{j_{1}}\wedge\cdots\wedge
e_{j_{q-p}}\right\rangle\widetilde{\Phi}_{p}\wedge\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}}\right\rangle\\\
=\left\langle
X^{q},\widetilde{\Phi}_{p}\wedge\frac{1}{\left(q-p\right)!}\left\langle\Psi_{q-p},e_{j_{1}}\wedge\cdots\wedge
e_{j_{q-p}}\right\rangle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{q-p}}\right\rangle\\\
=\left\langle
X^{q},\widetilde{\Phi}_{p}\wedge\Psi_{q-p}\right\rangle,\end{array}$

and the result is proved.

####  3.2.2 Right Contracted Product

###### Definition 12

The right contracted product of a multiform $\Phi$ with a multivector $X$ (or,
a multivector $X$ with a multiform $\Phi$) is the multiform
$\left|\Phi,X\right\rangle$ (respectively, the multivector
$\left|X,\Phi\right\rangle$) given by the following axioms:

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V\ddot{}$ with $n\geq p\geq 0:$

$\left|\Phi_{p},X^{p}\right\rangle=\left|X^{p},\Phi_{p}\right\rangle=\left\langle\widetilde{\Phi}_{p},X^{p}\right\rangle=\left\langle\Phi_{p},\widetilde{X}^{p}\right\rangle.$
(28)

For all $\Phi_{p}\in{\textstyle\bigwedge\nolimits^{p}}V^{\ast}$ and
$X^{q}\in{\textstyle\bigwedge\nolimits^{q}}V\ddot{}$ (or
$X^{p}\in{\textstyle\bigwedge\nolimits^{p}}V$ and
$\Phi_{q}\in{\textstyle\bigwedge\nolimits^{q}}V^{\ast}$) with $n\geq p>q\geq
0:$

$\displaystyle\left|\Phi_{p},X^{q}\right\rangle$
$\displaystyle=\frac{1}{(p-q)!}\left\langle\Phi_{p},e_{j_{1}}\wedge\cdots\wedge
e_{j_{p-q}}\wedge\widetilde{X}^{q}\right\rangle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{p-q}},$
(29) $\displaystyle\left|X^{p},\Phi_{q}\right\rangle$
$\displaystyle=\frac{1}{(p-q)!}\left\langle
X^{p},\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{p-q}}\wedge\widetilde{\Phi}_{q}\right\rangle
e_{j_{1}}\wedge\cdots\wedge e_{j_{p-q}},$ (30)

where $\left\\{e_{j},\varepsilon^{j}\right\\}$ is any pair of dual bases for
$V$ and $V^{\ast}$.

For all $\Phi\in{\textstyle\bigwedge}V^{\ast}$ and
$X\in{\textstyle\bigwedge}V:$ if $\Phi=\Phi_{0}+\Phi_{1}+\cdots+\Phi_{n}$ and
$X=X^{0}+X^{1}+\cdots+X^{n}$ it is:

$\displaystyle\left|\Phi,X\right\rangle$
$\displaystyle=\overset{n}{\underset{k=0}{{\textstyle\sum}}}\underset{j=0}{\overset{n-k}{{\textstyle\sum}}}\left|\Phi_{k+j},X^{j}\right\rangle,$
(31) $\displaystyle\left|X,\Phi\right\rangle$
$\displaystyle=\overset{n}{\underset{k=0}{{\textstyle\sum}}}\underset{j=0}{\overset{n-k}{{\textstyle\sum}}}\left|X^{k+j},\Phi_{j}\right\rangle.$
(32)

The right contracted product satisfies properties similar to the left
contracted product (see [4]).

###  3.3 Duality Adjoint of Extensors

In the Appendix D we briefly recall the extensor concept and the exterior
product of extensors associated with a real vector space $V$ of finite
dimension which is need for the following. Let
${\textstyle\bigwedge\nolimits^{\diamond}}V$ be any sum of homogeneous
subspaces of ${\textstyle\bigwedge}V.$ There exist $\mu$ integer numbers
$p_{1},\ldots,p_{\mu}$ with $0\leq p_{1}<\cdots<p_{\mu}\leq n$ such that
${\textstyle\bigwedge\nolimits^{\diamond}}V={\textstyle\bigwedge\nolimits^{p_{1}}}V+\cdots+{\textstyle\bigwedge\nolimits^{p_{\mu}}}V.$
Analogously, if ${\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}$ is any sum
of homogeneous subspaces of ${\textstyle\bigwedge}V^{\ast},$ then there exist
$\nu$ integer numbers $q_{1},\ldots,q_{\nu}$ with $0\leq
q_{1}<\cdots<q_{\nu}\leq n$ such that
${\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}={\textstyle\bigwedge\nolimits^{q_{1}}}V^{\ast}+\cdots+{\textstyle\bigwedge\nolimits^{q_{\nu}}}V^{\ast}.$

###### Definition 13

The linear mappings

${\textstyle\bigwedge}V\ni X\mapsto\left\langle
X\right\rangle^{{\textstyle\bigwedge\nolimits^{\diamond}}V}\in{\textstyle\bigwedge}V\text{
and
}{\textstyle\bigwedge}V^{\ast}\ni\Phi\mapsto\left\langle\Phi\right\rangle_{{\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}}\in{\textstyle\bigwedge}V^{\ast}$

such that if
${\textstyle\bigwedge\nolimits^{\diamond}}V={\textstyle\bigwedge\nolimits^{p_{1}}}V+\cdots+{\textstyle\bigwedge\nolimits^{p_{\mu}}}V$
and
${\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}={\textstyle\bigwedge\nolimits^{q_{1}}}V^{\ast}+\cdots+{\textstyle\bigwedge\nolimits^{q_{\nu}}}V^{\ast},$
then

$\left\langle
X\right\rangle^{{\textstyle\bigwedge\nolimits^{\diamond}}V}=\left\langle
X\right\rangle^{p_{1}}+\cdots+\left\langle X\right\rangle^{p_{\mu}}\text{ and
}\left\langle\Phi\right\rangle_{{\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}}=\left\langle\Phi\right\rangle_{q_{1}}+\cdots+\left\langle\Phi\right\rangle_{q_{\nu}}$
(33)

are called the __ ${\textstyle\bigwedge\nolimits^{\diamond}}V$_-part operator_
_for multivectors_ and
${\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}$_-part operator for
multiforms,_ respectively.

It should be evident that for all $X\in{\textstyle\bigwedge}V$ and
$\Phi\in{\textstyle\bigwedge}V^{\ast}:$

$\displaystyle\left\langle
X\right\rangle^{{\textstyle\bigwedge\nolimits^{k}}V}$
$\displaystyle=\left\langle X\right\rangle^{k},$ (34)
$\displaystyle\left\langle\Phi\right\rangle_{{\textstyle\bigwedge\nolimits^{k}}V^{\ast}}$
$\displaystyle=\left\langle\Phi\right\rangle_{k}.$ (35)

Thus, ${\textstyle\bigwedge\nolimits^{\diamond}}V$-part operator and
${\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast}$-part operator are the
generalizations of $\left\langle\left.{}\right.\right\rangle^{k}$ and
$\left\langle\left.{}\right.\right\rangle_{k}.$

###### Definition 14

Let $\tau$ be a multivector extensor of either one multivector variable or one
multiform variable. The _duality adjoint of_ $\tau$ is given by:

If $\tau\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V)$
(or, $\tau\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V)$),
then $\tau^{\bigtriangleup}\in
ext({\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast};$
${\textstyle\bigwedge\nolimits_{1}^{\diamond}}V^{\ast})$ (respectively,
$\tau^{\bigtriangleup}\in
ext({\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V)$)
defined by

$\displaystyle\tau^{\bigtriangleup}(\Phi)$
$\displaystyle=\left\langle\Phi,\tau(\left\langle
1\right\rangle^{{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V})\right\rangle$
$\displaystyle+\overset{n}{\underset{k=1}{{\textstyle\sum}}}\frac{1}{k!}\left\langle\Phi,\tau(\left\langle
e_{j_{1}}\wedge\cdots\wedge
e_{j_{k}}\right\rangle^{{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V})\right\rangle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{k}}$
(36)

for each $\Phi\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast}$
(respectively, for each
$\Phi\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast}$)

$\displaystyle\tau^{\bigtriangleup}(\Phi)$
$\displaystyle=\left\langle\Phi,\tau(\left\langle
1\right\rangle_{{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}})\right\rangle$
$\displaystyle+\overset{n}{\underset{k=1}{{\textstyle\sum}}}\frac{1}{k!}\left\langle\Phi,\tau(\left\langle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{k}}\right\rangle_{{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}})\right\rangle
e_{j_{1}}\wedge\cdots\wedge e_{j_{k}}$ (37)

The basic properties of the adjoint of multivector extensors are given by:

###### Proposition 15

Let $\tau\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V).$
For all $X\in{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V$ and
$\Phi\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast},$ it holds

$\left\langle\tau(X),\Phi\right\rangle=\left\langle
X,\tau^{\bigtriangleup}(\Phi)\right\rangle.$ (38)

Let $\tau\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V).$
For all $\Phi\in{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}$ and
$\Psi\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast},$ it holds

$\left\langle\tau(\Phi),\Psi\right\rangle=\left\langle\Phi,\tau^{\bigtriangleup}(\Psi)\right\rangle.$
(39)

Proof. We present only the proof of the property given by Eq.(38), the other
one is similar. First, observe that if
$X\in{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V,$ then there exists $\mu$
integer numbers $p_{1},\ldots,p_{\mu}$ with $0\leq p_{1}<\cdots<p_{\mu}\leq n$
such that $X=X^{p_{1}}+...+X^{p_{\mu}}$ with
$X^{p_{i}}\in{\textstyle\bigwedge\nolimits^{p_{i}}}V$, where the
${\textstyle\bigwedge\nolimits^{p_{i}}}V$ are homogeneous subspace of
${\textstyle\bigwedge}V,$ thus if $\tau\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V)$,
we have that

$\tau:{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V\rightarrow{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V\qquad\text{or\qquad}\tau:{\textstyle\bigwedge\nolimits^{p_{1}}}V+...+{\textstyle\bigwedge\nolimits^{p_{\mu}}}V\rightarrow{\textstyle\bigwedge\nolimits^{q_{1}}}V+...+{\textstyle\bigwedge\nolimits^{q_{\mu}}}V,$

where
${\textstyle\bigwedge\nolimits_{1}^{\diamond}}V={\textstyle\bigwedge\nolimits^{p_{1}}}V+...+{\textstyle\bigwedge\nolimits^{p_{\mu}}}V$
and
${\textstyle\bigwedge\nolimits_{2}^{\diamond}}V={\textstyle\bigwedge\nolimits^{q_{1}}}V+...+{\textstyle\bigwedge\nolimits^{q_{\mu}}}V$,
such that

$\tau\left(X\right)=\tau\left(X^{p_{1}}+...+X^{p_{\mu}}\right)=\tau\left(X^{p_{1}}\right)+...+\tau\left(X^{p_{\mu}}\right)\in{\textstyle\bigwedge\nolimits^{q_{1}}}V+...+{\textstyle\bigwedge\nolimits^{q_{\mu}}}V,$

thus, we can put

$\tau\left(X^{p_{i}}\right)\in{\textstyle\bigwedge\nolimits^{q_{i}}}V\qquad\text{or\qquad}\tau\left|{}_{{\textstyle\bigwedge\nolimits^{p_{i}}}V}\right.\equiv\tau_{p_{i}}\in
ext({\textstyle\bigwedge\nolimits^{p_{i}}}V;{\textstyle\bigwedge\nolimits^{q_{i}}}V).$

Now, with the above observation, we have

$\begin{array}[c]{ll}\left\langle\tau(X),\Phi\right\rangle&=\left\langle\tau\left(X^{p_{1}}+...+X^{p_{\mu}}\right),\Phi_{q_{1}}+...+\Phi_{q_{\mu}}\right\rangle\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
&=\left\langle\tau_{p_{1}}\left(X^{p_{1}}\right)+...+\tau_{p_{\mu}}\left(X^{p_{\mu}}\right),\Phi_{q_{1}}+...+\Phi_{q_{\mu}}\right\rangle\vskip
6.0pt plus 2.0pt minus 2.0pt\\\
&={\textstyle\sum\limits_{i,j}}\left\langle\tau_{p_{i}}(X^{p_{i}}),\Phi^{q_{j}}\right\rangle,\end{array}$
(40)

but, by axiom given in Eq.(11), we have that

$\left\langle\tau_{p_{i}}(X^{p_{i}}),\Phi^{q_{j}}\right\rangle\left\\{\begin{array}[c]{l}=0\qquad\text{if
\ }p_{i}\neq q_{j}\\\ \neq 0\qquad\text{if \
}p_{i}=q_{j}=s_{l}\end{array}\right.,$

and from Eq. (40), we can write

$\left\langle\tau(X),\Phi\right\rangle=\left\langle\tau_{s_{1}}(X^{s_{1}}),\Phi^{s_{1}}\right\rangle+...+\left\langle\tau_{s_{\mu}}(X^{s_{\mu}}),\Phi^{s_{\mu}}\right\rangle.$
(41)

Now, if we see $\left\langle\tau_{s_{l}}(X^{s_{l}}),\Phi^{s_{l}}\right\rangle$
as a scalar product of $s_{l}$-vectors, then we have

$\left\langle\tau_{s_{l}}(X^{s_{l}}),\Phi^{s_{l}}\right\rangle=\left\langle
X^{s_{l}},\tau_{s_{l}}^{\bigtriangleup}\Phi^{s_{l}}\right\rangle,$ (42)

and from Eqs. (41), (42), and taking into account Eq. (38) we have

$\begin{array}[c]{ll}\left\langle\tau(X),\Phi\right\rangle&=\left\langle\tau_{s_{1}}(X^{s_{1}}),\Phi^{s_{1}}\right\rangle+...+\left\langle\tau_{s_{\mu}}(X^{s_{\mu}}),\Phi^{s_{\mu}}\right\rangle\\\
&=\left\langle
X^{s_{1}},\tau_{s_{1}}^{\bigtriangleup}\Phi^{s_{1}}\right\rangle+...+\left\langle
X^{s_{\mu}},\tau_{s_{\mu}}^{\bigtriangleup}\Phi^{s_{\mu}}\right\rangle\\\
&=\left\langle
X^{s_{1}}+...+X^{s_{\mu}},\tau_{s_{1}}^{\bigtriangleup}\Phi^{s_{1}}+...+\tau_{s_{\mu}}^{\bigtriangleup}\Phi^{s_{\mu}}\right\rangle\\\
&=\left\langle X,\tau^{\bigtriangleup}\Phi\right\rangle,\end{array}$

and the result is proved.

###### Definition 16

Let $\sigma$ be a multiform extensor of either one multivector variable or one
multiform variable. If $\sigma\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast})$
(or, $\sigma\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast})$),
then The _duality adjoint operator_
$\left(\left.{}\right.\right)^{\bigtriangleup}$ of $\sigma^{\bigtriangleup}\in
ext({\textstyle\bigwedge\nolimits_{2}^{\diamond}}V;$
${\textstyle\bigwedge\nolimits_{1}^{\diamond}}V^{\ast})$ (respectively,
$\sigma^{\bigtriangleup}\in
ext({\textstyle\bigwedge\nolimits_{4}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V)$)
is given by

$\displaystyle\sigma^{\bigtriangleup}(X)$ $\displaystyle=\left\langle
X,\sigma(\left\langle
1\right\rangle^{{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V})\right\rangle$
$\displaystyle+\overset{n}{\underset{k=1}{{\textstyle\sum}}}\frac{1}{k!}\left\langle
X,\sigma(\left\langle e_{j_{1}}\wedge\cdots\wedge
e_{j_{k}}\right\rangle^{{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V})\right\rangle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{k}}$
(43)

for each $X\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V$ (respectively,

$\displaystyle\sigma^{\bigtriangleup}(X)$ $\displaystyle=\left\langle
X,\sigma(\left\langle
1\right\rangle_{{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}})\right\rangle$
$\displaystyle+\overset{n}{\underset{k=1}{{\textstyle\sum}}}\frac{1}{k!}\left\langle
X,\sigma(\left\langle\varepsilon^{j_{1}}\wedge\cdots\wedge\varepsilon^{j_{k}}\right\rangle_{{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}})\right\rangle
e_{j_{1}}\wedge\cdots\wedge e_{j_{k}}$ (44)

for each $X\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V$).

The basic properties for the adjoint of multiform extensors are given by:

Let $\sigma\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V;{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast})$.
For all $X\in{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V$ and
$Y\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V,$ it holds

$\left\langle\sigma(X),Y\right\rangle=\left\langle
X,\sigma^{\bigtriangleup}(Y)\right\rangle.$ (45)

Let $\sigma\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast}).$
For all $\Phi\in{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V^{\ast}$ and
$X\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V,$ it holds

$\left\langle\sigma(\Phi),X\right\rangle=\left\langle\Phi,\sigma^{\bigtriangleup}(X)\right\rangle.$
(46)

###  3.4 Extended Operators on Extensors

Let $\lambda$ be an invertible linear operator on $V$. As we saw in appendix B
and C, $\lambda$ can be extended by considering two different procedures, non-
equivalent, denoted by $\underline{\lambda}$ and $\underset{\smile}{\gamma}$
both mapping multivectors over $V$ into multivectors over $V.$ However, it is
possible to extend the action of $\underline{\lambda}$ and
$\underset{\smile}{\gamma}$ in such a way that they map multivector extensors
over $V$ into multivector extensors over $V$.

####  3.4.1 Exterior Power Extension Operators on Extensors

###### Definition 17

Let $\lambda$ be an invertible linear operator on $V$. The exterior power
extension of a vector operator $\lambda$ on multivector extensors is the
linear mapping

$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)\ni\tau\mapsto\underline{\lambda}\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)$

such that

$\underline{\lambda}\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})=\underline{\lambda}\circ\tau(\underline{\lambda}^{-1}(X_{1}),\ldots,\underline{\lambda}^{-1}(X_{k}),\underline{\lambda}^{\bigtriangleup}(\Phi^{1}),\ldots,\underline{\lambda}^{\bigtriangleup}(\Phi^{l}))$
(47)

for each $X_{1},\ldots,X_{k}\in{\textstyle\bigwedge}V$ and
$\Phi^{1},\ldots,\Phi^{l}\in{\textstyle\bigwedge}V^{\ast}$.

It that way $\underline{\lambda}$ can be thought as a linear multivector
extensor operator.

For instance, for
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{1}^{0}(V)$ the above
definition means

$\underline{\lambda}\tau=\underline{\lambda}\circ\tau\circ\underline{\lambda}^{-1},$
(48)

and for $\tau\in\left.\overset{\left.{}\right.}{ext}\right._{0}^{1}(V)$ it
yields

$\underline{\lambda}\tau=\underline{\lambda}\circ\tau\circ\underline{\lambda}^{\bigtriangleup}.$
(49)

Let $\lambda$ be an invertible linear operator on $V^{\ast}.$ Analogously to
the above case, it is possible to extend the action of $\underline{\lambda}$
in such a way to get a linear operator on
$\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V).$ We define

$\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)\ni\tau\mapsto\underline{\lambda}\tau\in\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)$

such that

$\underline{\lambda}\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})=\underline{\lambda}\circ\tau(\underline{\lambda}^{\bigtriangleup}(X_{1}),\ldots,\underline{\lambda}^{\bigtriangleup}(X_{k}),\underline{\lambda}^{-1}(\Phi^{1}),\ldots,\underline{\lambda}^{-1}(\Phi^{l}))$
(50)

for each $X_{1},\ldots,X_{k}\in{\textstyle\bigwedge}V$ and
$\Phi^{1},\ldots,\Phi^{l}\in{\textstyle\bigwedge}V^{\ast}$.

For instance, for
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{1}^{0}(V)$ the above
definition yields

$\underline{\lambda}\tau=\underline{\lambda}\circ\tau\circ\underline{\lambda}^{\bigtriangleup},$
(51)

and for $\tau\in\left.\overset{\left.{}\right.}{ext}\right._{0}^{1}(V)$ it
implies that

$\underline{\lambda}\tau=\underline{\lambda}\circ\tau\circ\underline{\lambda}^{-1}.$
(52)

We give some of the properties of the action of the exterior power extension
of a vector operator $\lambda$ on multivector extensors.

###### Proposition 18

For all $\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\underline{\lambda}(\tau\wedge\sigma)=(\underline{\lambda}\tau)\wedge(\underline{\lambda}\sigma).$
(53)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\displaystyle\underline{\lambda}\left\langle\tau,\sigma\right\rangle$
$\displaystyle=\left\langle\underline{\lambda}^{-\bigtriangleup}\tau,\underline{\lambda}\sigma\right\rangle,$
(54) $\displaystyle\underline{\lambda}\left\langle\tau,\sigma\right|$
$\displaystyle=\left\langle\underline{\lambda}^{-\bigtriangleup}\tau,\underline{\lambda}\sigma\right|,$
(55) $\displaystyle\underline{\lambda}\left|\sigma,\tau\right\rangle$
$\displaystyle=\left|\underline{\lambda}\sigma,\underline{\lambda}^{-\bigtriangleup}\tau\right\rangle.$
(56)

We present the proof of the property given by Eq.(55), the other proof are
similar.

Proof. Without any loss of generality, we give the proof for the particular
case where $\tau\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V,{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast})$
and $\sigma\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V,{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits^{\diamond}}V)$.
Take $X\in{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V,$
$Y\in{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V$ and
$\Phi\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast},$
$\Psi\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast}.$ A
straightforward calculation, using Eq.(47), Eq.(216), Eq.(188) and Eq.(50),
gives

$\displaystyle\underline{\lambda}\left\langle\tau,\sigma\right|(X,Y,\Phi,\Psi)$
$\displaystyle=\underline{\lambda}\circ\left\langle\tau,\sigma\right|(\underline{\lambda}^{-1}(X),\underline{\lambda}^{-1}(Y),\underline{\lambda}^{\bigtriangleup}(\Phi),\underline{\lambda}^{\bigtriangleup}(\Psi))$
$\displaystyle=\underline{\lambda}\left\langle\tau(\underline{\lambda}^{-1}(X),\underline{\lambda}^{\bigtriangleup}(\Phi)),\sigma(\underline{\lambda}^{-1}(Y),\underline{\lambda}^{\bigtriangleup}(\Psi))\right|$
$\displaystyle=\left\langle\underline{\lambda}^{-\bigtriangleup}\circ\tau(\underline{\lambda}^{-1}(X),\underline{\lambda}^{\bigtriangleup}(\Phi)),\underline{\lambda}\circ\sigma(\underline{\lambda}^{-1}(Y),\underline{\lambda}^{\bigtriangleup}(\Psi))\right|$
$\displaystyle=\left\langle\underline{\lambda}^{-\bigtriangleup}\tau(X,\Phi),\underline{\lambda}\sigma(Y,\Psi)\right|$
$\displaystyle=\left\langle\underline{\lambda}^{-\bigtriangleup}\tau,\underline{\lambda}\sigma\right|(X,Y,\Phi,\Psi),$

whence, the expected result follows.

We present now some of the properties of the action of the exterior power
extension of a form operator $\lambda$ on multiform extensors.

###### Proposition 19

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\ast}{ext}\right._{r}^{s}(V):$

$\underline{\lambda}(\tau\wedge\sigma)=(\underline{\lambda}\tau)\wedge\underline{\lambda}(\sigma).$
(57)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\displaystyle\underline{\lambda}\left\langle\tau,\sigma\right\rangle$
$\displaystyle=\left\langle\underline{\lambda}\tau,\underline{\lambda}^{-\bigtriangleup}\sigma\right\rangle,$
(58) $\displaystyle\underline{\lambda}\left\langle\tau,\sigma\right|$
$\displaystyle=\left\langle\underline{\lambda}\tau,\underline{\lambda}^{-\bigtriangleup}\sigma\right|,$
(59) $\displaystyle\underline{\lambda}\left|\sigma,\tau\right\rangle$
$\displaystyle=\left|\underline{\lambda}^{-\bigtriangleup}\sigma,\underline{\lambda}\tau\right\rangle.$
(60)

The proofs are similar to the one of the previous proposition.

####  3.4.2 Action of Contracted Extension Operators on Extensors

###### Definition 20

Let $\gamma$ be a linear operator on $V$. Its contracted extension operator is
an operator $\underset{\smile}{\gamma}$ which maps map multivector extensors
over $V$ into multivector extensors over $V$. We have

$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)\ni\tau\mapsto\underset{\smile}{\gamma}\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)$

such that

$\displaystyle\underset{\smile}{\gamma}\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle=\underset{\smile}{\gamma}\circ\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle-\tau(\underset{\smile}{\gamma}(X_{1}),\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle\cdots-\tau(X_{1},\ldots,\underset{\smile}{\gamma}(X_{k}),\Phi^{1},\ldots,\Phi^{l})$
(61)
$\displaystyle+\tau(X_{1},\ldots,X_{k},\underset{\smile}{\gamma}^{\bigtriangleup}(\Phi^{1}),\ldots,\Phi^{l})$
$\displaystyle\cdots+\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\underset{\smile}{\gamma}^{\bigtriangleup}(\Phi^{l})$

for each $X_{1},\ldots,X_{k}\in{\textstyle\bigwedge}V$ and
$\Phi^{1},\ldots,\Phi^{l}\in{\textstyle\bigwedge}V^{\ast}$.

Thus, we can think of $\underset{\smile}{\gamma}$ as a linear _multivector
extensor operator_. The definitions of $\underset{\smile}{\gamma}$ for
${\textstyle\bigwedge}V$ and ${\textstyle\bigwedge}V^{\ast}$ are given by
Eq.(193) and Eq.(198) respectively.

For instance, for
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{1}^{0}(V)$ the above
definition gives

$\underset{\smile}{\gamma}\tau=\underset{\smile}{\gamma}\circ\tau-\tau\circ\underset{\smile}{\gamma}=\left[\underset{\smile}{\gamma},\tau\right],$
(62)

and for $\tau\in\left.\overset{\left.{}\right.}{ext}\right._{0}^{1}(V)$ it
yields

$\underset{\smile}{\gamma}\tau=\underset{\smile}{\gamma}\circ\tau+\tau\circ\underset{\smile}{\gamma}^{\bigtriangleup}.$
(63)

###### Definition 21

Let $\gamma$ be an invertible linear operator on $V^{\ast}$. Analogously to
the above case, it is possible to generalize the action of
$\underset{\smile}{\gamma}$ in such a way to get a linear operator on
$\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)$. We have

$\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)\ni\tau\mapsto\underset{\smile}{\gamma}\tau\in\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)$

such that

$\displaystyle\underset{\smile}{\gamma}\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle=\underset{\smile}{\gamma}\circ\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle+\tau(\underset{\smile}{\gamma}^{\bigtriangleup}(X_{1}),\ldots,X_{k},\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle\cdots+\tau(X_{1},\ldots,\underset{\smile}{\gamma}^{\bigtriangleup}(X_{k}),\Phi^{1},\ldots,\Phi^{l})$
$\displaystyle-\tau(X_{1},\ldots,X_{k},\underset{\smile}{\gamma}(\Phi^{1}),\ldots,\Phi^{l})$
$\displaystyle\cdots-\tau(X_{1},\ldots,X_{k},\Phi^{1},\ldots,\underset{\smile}{\gamma}(\Phi^{l})$
(64)

for each $X_{1},\ldots,X_{k}\in{\textstyle\bigwedge}V$ and
$\Phi^{1},\ldots,\Phi^{l}\in{\textstyle\bigwedge}V^{\ast}.$

For instance, for $\tau\in\left.\overset{\ast}{ext}\right._{1}^{0}(V)$ the
above definition yields

$\underset{\smile}{\gamma}\tau=\underset{\smile}{\gamma}\circ\tau+\tau\circ\underset{\smile}{\gamma}^{\bigtriangleup},$
(65)

and for $\tau\in\left.\overset{\left.\ast\right.}{ext}\right._{0}^{1}(V)$ it
holds

$\underset{\smile}{\gamma}\tau=\underset{\smile}{\gamma}\circ\tau-\tau\circ\underset{\smile}{\gamma}=\left[\underset{\smile}{\gamma},\tau\right].$
(66)

We present some of the main properties of the action of the contracted
extension operator of a vector operator $\gamma$ on multivector extensors.

###### Proposition 22

For all $\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\underset{\smile}{\gamma}(\tau\wedge\sigma)=(\underset{\smile}{\gamma}\tau)\wedge\sigma+\tau\wedge(\underset{\smile}{\gamma}\sigma).$
(67)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\displaystyle\underset{\smile}{\gamma}\left\langle\tau,\sigma\right\rangle$
$\displaystyle=-\left\langle\underset{\smile}{\gamma}^{\bigtriangleup}\tau,\sigma\right\rangle+\left\langle\tau,\underset{\smile}{\gamma}\sigma\right\rangle,$
(68) $\displaystyle\underset{\smile}{\gamma}\left\langle\tau,\sigma\right|$
$\displaystyle=-\left\langle\underset{\smile}{\gamma}^{\bigtriangleup}\tau,\sigma\right|+\left\langle\tau,\underset{\smile}{\gamma}\sigma\right|,$
(69) $\displaystyle\underset{\smile}{\gamma}\left|\sigma,\tau\right\rangle$
$\displaystyle=\left|\underset{\smile}{\gamma}\sigma,\tau\right\rangle-\left|\sigma,\underset{\smile}{\gamma}^{\bigtriangleup}\tau\right\rangle.$
(70)

Proof. We present only the proof for the property given by Eq. (67), the
others are similar. Without any loss of generality, we give the proof for the
particular case where $\tau\in
ext({\textstyle\bigwedge\nolimits_{1}^{\diamond}}V,{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits^{\diamond}}V^{\ast})$
and $\sigma\in
ext({\textstyle\bigwedge\nolimits_{3}^{\diamond}}V,{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast};{\textstyle\bigwedge\nolimits^{\diamond}}V)$.
Take $X\in{\textstyle\bigwedge\nolimits_{1}^{\diamond}}V,$
$Y\in{\textstyle\bigwedge\nolimits_{3}^{\diamond}}V$ and
$\Phi\in{\textstyle\bigwedge\nolimits_{2}^{\diamond}}V^{\ast},$
$\Psi\in{\textstyle\bigwedge\nolimits_{4}^{\diamond}}V^{\ast}$. By using the
definition of $\underset{\smile}{\gamma}$ , we have

$\begin{array}[c]{ll}\underset{\smile}{\gamma}(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)&=\underset{\smile}{\gamma}\circ(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)-(\tau\wedge\sigma)\left(\underset{\smile}{\gamma}X,Y,\Phi,\Psi\right)\\\
&-(\tau\wedge\sigma)\left(X,\underset{\smile}{\gamma}Y,\Phi,\Psi\right)+(\tau\wedge\sigma)\left(X,Y,\underset{\smile}{\gamma}\Phi,\Psi\right)\\\
&+(\tau\wedge\sigma)\left(X,Y,\Phi,\underset{\smile}{\gamma}\Psi\right).\end{array}$
(71)

Now, using the property (197) we can write the first term of right side of the
Eq. (71) as

$\underset{\smile}{\gamma}\circ(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)=\underset{\smile}{\gamma}\circ\tau\left(X,\Phi\right)\wedge\sigma\left(Y,\Psi\right)+\tau\left(X,\Phi\right)\wedge\underset{\smile}{\gamma}\circ\sigma\left(Y,\Psi\right),$

and remembering that

$(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)=\tau\left(X,\Phi\right)\wedge\sigma\left(Y,\Psi\right),$

the Eq. (71) can be written as

$\begin{array}[c]{ll}\underset{\smile}{\gamma}(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)&=\left[\underset{\smile}{\gamma}\circ\tau\left(X,\Phi\right)-\tau\left(\underset{\smile}{\gamma}X,\Phi\right)+\tau\left(X,\underset{\smile}{\gamma^{\bigtriangleup}}\Phi\right)\right]\wedge\sigma\left(Y,\Psi\right)\\\
&+\tau\left(X,\Phi\right)\wedge\left[\underset{\smile}{\gamma}\circ\sigma\left(Y,\Psi\right)-\sigma\left(\underset{\smile}{\gamma}Y,\Psi\right)+\sigma\left(Y,\underset{\smile}{\gamma^{\bigtriangleup}}\Psi\right)\right]\end{array}$

or

$\begin{array}[c]{ll}\underset{\smile}{\gamma}(\tau\wedge\sigma)\left(X,Y,\Phi,\Psi\right)&=\left[\left(\underset{\smile}{\gamma}\tau\right)\left(X,\Phi\right)\right]\wedge\sigma\left(Y,\Psi\right)+\tau\left(X,\Phi\right)\wedge\left[\left(\underset{\smile}{\gamma}\sigma\right)\left(Y,\Psi\right)\right]\\\
&=\left(\underset{\smile}{\gamma}\tau\wedge\sigma+\tau\wedge\underset{\smile}{\gamma}\sigma\right)\left(X,Y,\Phi,\Psi\right),\end{array}$

and the property is proved.

We present some properties for the action of the contacted extension operator
of a form operator $\gamma$ on multiform extensors.

###### Proposition 23

For all $\tau\in\left.\overset{\left.\ast\right.}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.\ast\right.}{ext}\right._{r}^{s}(V):$

$\underset{\smile}{\gamma}(\tau\wedge\sigma)=(\underset{\smile}{\gamma}\tau)\wedge\sigma+\tau\wedge(\underset{\smile}{\gamma}\sigma).$
(72)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}(V)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}(V):$

$\displaystyle\underset{\smile}{\gamma}\left\langle\tau,\sigma\right\rangle$
$\displaystyle=\left\langle\underset{\smile}{\gamma}\tau,\sigma\right\rangle-\left\langle\tau,\underset{\smile}{\gamma}^{\bigtriangleup}\sigma\right\rangle,$
(73) $\displaystyle\underset{\smile}{\gamma}\left\langle\tau,\sigma\right|$
$\displaystyle=\left\langle\underset{\smile}{\gamma}\tau,\sigma\right|-\left\langle\tau,\underset{\smile}{\gamma}^{\bigtriangleup}\sigma\right|,$
(74) $\displaystyle\underset{\smile}{\gamma}\left|\sigma,\tau\right\rangle$
$\displaystyle=-\left|\underset{\smile}{\gamma}^{\bigtriangleup}\sigma,\tau\right\rangle+\left|\sigma,\underset{\smile}{\gamma}\tau\right\rangle.$
(75)

Proofs are analogous to the ones of the previous proposition.

###  3.5 Multivector and Multiform Fields

Let $U$ be an open set of a smooth manifold $M$ (i.e., $U\subseteq M$). The
set of smooth111Smooth in this paper means,
$\mathcal{C}^{\infty}$-differentiable or at least enough differentiable in
order for our statements to hold. scalar fields on $U,$ as well-known, has a
natural structure of _ring_ (_with identity_), and it will be denoted by
$\mathcal{S}(U).$ The set of smooth vector fields on $U,$ as well-known, have
natural structure of _modules over_ $\mathcal{S}(U).$ It will be denoted by
$\mathcal{V}(U)$. The set of smooth form fields on $U$ _can be identified_
with the _dual module_ for $\mathcal{V}(U)$. It could be denoted by
$\mathcal{V}^{\ast}(U)$.

Let $M$ be a $n$-dimensional differentiable manifold and let $U\subset M$ be
an open set. A $k$-vector mapping

$X^{k}:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}{\textstyle\bigwedge\nolimits^{k}}T_{p}^{\ast}M,$

such that for each $p\in U,$
$X_{(p)}^{k}\in{\textstyle\bigwedge\nolimits^{k}}T_{p}^{\ast}M$ is called a
$k$_-vector field on_ $U.$

Such $X^{k}$ with $1\leq k\leq n$ is said to be a smooth $k$-vector field on
$U,$ if and only if, for all
$\omega^{1},\ldots,\omega^{k}\in\mathcal{V}^{\ast}(U)$, the scalar mapping
defined by

$U\ni p\mapsto
X_{(p)}^{k}(\omega_{(p)}^{1},\ldots,\omega_{(p)}^{k})\in\mathbb{R}$ (76)

is a smooth scalar field on $U$.

A multivector mapping

$X:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}{\textstyle\bigwedge}T_{p}^{\ast}M,$

such that for each $p\in U$, $X_{(p)}\in{\textstyle\bigwedge}T_{p}^{\ast}M$ is
called a _multivector field on_ $U.$

Any multivector at $p\in M$ can be written as a sum of $k$-vectors (i.e.,
homogeneous multivectors of degree $k$) at $p\in M,$ with $k$ running from
$k=0$ to $k=n$, i.e., there exist exactly $n+1$ homogeneous multivector of
degree $k$ fields on $U,$ conveniently denote by $X^{0},X^{1},\ldots,X^{n},$
such that for every $p\in U,$

$X_{(p)}=X_{(p)}^{0}+X_{(p)}^{1}+\cdots+X_{(p)}^{n}.$ (77)

We say that $X$ is a smooth multivector field on $U$ when each of one of the
$X^{0},X^{1},\ldots,X^{n}$ is a smooth $k$-vector field on $U.$

We emphasize that according with the definitions of smoothness as given above,
a smooth $k$-vector field on $U$ _can be identified_ to a $k$-vector over
$\mathcal{V}(U)$, and a smooth multivector field on $U$ _can be seen properly_
as a multivector over $\mathcal{V}(U).$ Thus, the set of smooth $k$-vector
fields on $U$ may be denoted by
${\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U)$, and the set of smooth
multivector fields on $U$ may be denoted by
${\textstyle\bigwedge}\mathcal{V}(U)$.

A $k$-form mapping

$\Phi_{k}:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}{\textstyle\bigwedge\nolimits^{k}}T_{p}^{\ast}M$

such that for each $p\in U$,
$\Phi_{k(p)}\in{\textstyle\bigwedge\nolimits^{k}}T_{p}^{\ast}M$ is called a
$k$_-form field on_ $U.$

Such $\Phi_{k}$ with $1\leq k\leq n$ is said to be a smooth $k$-form field on
$U,$ if and only if, for all $v_{1},\ldots,v_{k}\in\mathcal{V}(U)$, the scalar
mapping defined by

$U\ni p\mapsto\Phi_{k(p)}(v_{1(p)},\ldots,v_{k(p)})\in\mathbb{R}$ (78)

is a smooth scalar field on $U$.

A multiform mapping

$\Phi:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}{\textstyle\bigwedge}T_{p}^{\ast}M$

such that for each $p\in U$, $\Phi_{(p)}\in{\textstyle\bigwedge}T_{p}^{\ast}M$
is called a _multiform field on_ $U.$

Any multiform at $p\in M$ can be written (see, e.g.,[27]) as a sum of
$k$-forms (i.e., homogeneous multiforms of degree $k$) at $p\in M$ with $k$
running from $k=0$ to $k=n.$ It follows that there exist exactly $n+1$
homogeneous multiform of degree $k$ fields on $U,$ named as
$\Phi_{0},\Phi_{1},\ldots,\Phi_{n}$ such that

$\Phi_{(p)}=\Phi_{0(p)}+\Phi_{1(p)}+\cdots+\Phi_{n(p)}$ (79)

for every $p\in U.$

We say that $\Phi$ is a smooth multiform field on $U,$ if and only if, each of
$\Phi_{0},\Phi_{1},\ldots,\Phi_{n}$ is just a smooth $k$-form field on $U.$

Note that according with the definitions of smoothness as given above, a
smooth $k$-form field on $U$ _can be identified_ with a $k$-form over
$\mathcal{V}(U),$ and a smooth multiform field on $U$ _can be seen_ as a
multiform over $\mathcal{V}(U).$ Thus, the set of smooth $k$-form fields on
$U$ will be denoted by
${\textstyle\bigwedge\nolimits^{k}}\mathcal{V}^{\ast}(U),$ and the set of
smooth multiform fields on $U$ will be denoted by
${\textstyle\bigwedge}\mathcal{V}^{\ast}(U).$

####  3.5.1 Algebra of Multivector and Multiform Fields

We recall first the module (over a ring) structure operations of the set of
smooth multivector fields on $U$ and of the set of smooth multiform fields on
$U$. We recall also the concept of the exterior product both of smooth
multivector fields as well as of smooth multiform fields on $U$ and we present
the definitions of the duality products of a given smooth multivector fields
on $U$ by a smooth multiform field on $U.$

The addition of multivector fields $X$ and $Y,$ or multiform fields $\Phi$ and
$\Psi,$ is defined by

$\displaystyle\left(X+Y\right)_{(p)}$ $\displaystyle=X_{(p)}+Y_{(p)},$ (80)
$\displaystyle\left(\Phi+\Psi\right)_{(p)}$
$\displaystyle=\Phi_{(p)}+\Psi_{(p)},$ (81)

for every $p\in U.$

The scalar multiplication of a multivector field $X,$ or a multiform field
$\Phi,$ by a scalar field $f,$ is defined by

$\displaystyle\left(fX\right)_{(p)}$ $\displaystyle=f(p)X_{(p)},$ (82)
$\displaystyle\left(f\Phi\right)_{(p)}$ $\displaystyle=f(p)\Phi_{(p)},$ (83)

for every $p\in U.$

The exterior product of multivector fields $X$ and $Y,$ and the exterior
product of multiform fields $\Phi$ and $\Psi,$ are defined by

$\displaystyle\left(X\wedge Y\right)_{(p)}$ $\displaystyle=X_{(p)}\wedge
Y_{(p)},$ (84) $\displaystyle\left(\Phi\wedge\Psi\right)_{(p)}$
$\displaystyle=\Phi_{(p)}\wedge\Psi_{(p)},$ (85)

for every $p\in U.$

Each module, of either the smooth multivector fields on $U$, or the smooth
multiform fields on $U$, endowed with the respective exterior product has a
natural structure of associative algebra. They are called _the exterior
algebras of multivector and multiform fields on_ $U.$

The duality scalar product of a multiform field $\Phi$ with a multivector
field $X$ is (see algebraic details in [4]) the scalar field
$\left\langle\Phi,X\right\rangle$ defined by

$\left\langle\Phi,X\right\rangle(p)=\left\langle\Phi_{(p)},X_{(p)}\right\rangle,$
(86)

for every $p\in U.$

The duality left contracted product of a multiform field $\Phi$ with a
multivector field $X$ (or, a multivector field $X$ with a multiform field
$\Phi$) is the multivector field $\left\langle\Phi,X\right|$ (respectively,
the multiform field $\left\langle X,\Phi\right|$) defined by

$\displaystyle\left\langle\Phi,X\right|_{(p)}$
$\displaystyle=\left\langle\Phi_{(p)},X_{(p)}\right|,$ (87)
$\displaystyle\left\langle X,\Phi\right|_{(p)}$ $\displaystyle=\left\langle
X_{(p)},\Phi_{(p)}\right|,$ (88)

for every $p\in U.$

The duality right contracted product of a multiform field $\Phi$ with a
multivector field $X$ (or, a multivector field $X$ with a multiform field
$\Phi$) is the multiform field $\left|\Phi,X\right\rangle$ (respectively, the
multivector field $\left|X,\Phi\right\rangle$) defined by

$\displaystyle\left|\Phi,X\right\rangle_{(p)}$
$\displaystyle=\left|\Phi_{(p)},X_{(p)}\right\rangle,$ (89)
$\displaystyle\left|X,\Phi\right\rangle_{(p)}$
$\displaystyle=\left|X_{(p)},\Phi_{(p)}\right\rangle,$ (90)

for every $p\in U.$

Each duality contracted product of smooth multivector fields on $U$ with
smooth multiform fields on $U$ yields a natural structure of non-associative
algebra.

###  3.6 Extensor Fields

Let $T_{p}M$ the set of all tangent vectors to $M$ at $p.$ A multivector
extensor mapping

$\tau:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(T_{p}M)$

such that for each $p\in U$,
$\tau_{(p)}\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(T_{p}M)$ is
called a _multivector extensor field of_ $k$_multivector and_ $l$_multiform
variables on_ $U.$

A multiform extensor mapping

$\upsilon:U\longrightarrow\underset{p\in
U}{{\textstyle\bigcup}}\left.\overset{\ast}{ext}\right._{k}^{l}(T_{p}M)$

such that for each $p\in U$,
$\upsilon_{(p)}\in\left.\overset{\ast}{ext}\right._{k}^{l}(T_{p}M)$ is called
a _multiform_ extensor _ field of _$k$_multivector and_ $l$_multiform
variables on_ $U.$

In the above formulas,
$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(T_{p}M)$ is a short
notation for the space of multivector extensors of $k$ multivector and $l$
multiform variables over $T_{p}M,$ i.e., for each $p\in U:$

$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}(T_{p}M):=ext({\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}M,\ldots,{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}T_{p}M,{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}^{\star}M,\ldots,{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}T_{p}^{\star}M;{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}M),$

and $\left.\overset{\ast}{ext}\right._{k}^{l}(T_{p}M)$ is a short notation for
the space of multiform extensors of $k$ multivector and $l$ multiform
variables over $T_{p}M,$ i.e., for each $p\in U:$

$\left.\overset{\ast}{ext}\right._{k}^{l}(T_{p}M):=ext({\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}M,\ldots,{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}T_{p}M,{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}^{\star}M,\ldots,{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}T_{p}^{\star}M;{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}^{\star}M),$

where $T_{p}^{\star}M$ is the dual space of $T_{p}M.$

Let us denote the smooth multivector fields: $U\ni p\mapsto
X_{1(p)}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}M,\ldots$,
$U\ni p\mapsto
X_{k(p)}\in{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}T_{p}M$, and $U\ni
p\mapsto X_{(p)}\in{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}M$,
respectively by
${\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}(U),\ldots$,
${\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}\mathcal{V}(U)$, and
${\displaystyle\bigwedge\nolimits^{\Diamond}}\mathcal{V}(U)$. Let us denote
the smooth multiform fields: $U\ni
p\mapsto\Phi_{(p)}^{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}T_{p}^{\ast}M,\ldots$,
$U\ni
p\mapsto\Phi_{(p)}^{l}\in{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}T_{p}^{\ast}M$,
and $U\ni
p\mapsto\Phi_{(p)}^{l}\in{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}^{\ast}M$,
by ${\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}^{\ast}(U),$
$\ldots,$
${\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}\mathcal{V}^{\ast}(U)$ and
${\displaystyle\bigwedge\nolimits^{\Diamond}}\mathcal{V}^{\ast}(U)$.

Such a multivector extensor field $\tau$ will be said to be smooth, if and
only if, for all
$X_{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}(U),\ldots,X_{k}\in{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}\mathcal{V}(U),$
and for all
$\Phi^{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}^{\ast}(U),\ldots,\Phi^{l}\in{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}\mathcal{V}^{\star}(U)$,
the multivector mapping defined by

$U\ni
p\mapsto\tau_{(p)}(X_{1(p)},\ldots,X_{k(p)},\Phi_{(p)}^{1},\ldots,\Phi_{(p)}^{l})\in{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}M$
(91)

is a smooth multivector field on $U,$ (i.e., an object living on
${\displaystyle\bigwedge\nolimits^{\Diamond}}\mathcal{V}(U)$).

Such a multiform extensor field $\upsilon$ will be said to be smooth , if and
only if, for all
$X_{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}(U),\ldots,X_{k}\in{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}\mathcal{V}(U),$
and for all
$\Phi^{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}^{\ast}(U),\ldots,\Phi^{l}\in{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}\mathcal{V}^{\ast}(U)$,
the multiform mapping defined by

$U\ni
p\mapsto\upsilon_{(p)}(X_{1(p)},\ldots,X_{k(p)},\Phi_{(p)}^{1},\ldots,\Phi_{(p)}^{l})\in{\displaystyle\bigwedge\nolimits^{\Diamond}}T_{p}^{\star}M$
(92)

is a smooth multiform field on $U,$ (i.e., an object living on
${\displaystyle\bigwedge^{\Diamond}}\mathcal{V}^{\star}(U)$).

We emphasize222A _short name_ for a multivector (or, multiform) extensor of
$k$ multivector and $l$ multiform variables could be: a $\binom{l}{k}$
multivector (respectively, multiform) extensor. that according with the
definitions of smoothness as given above, a smooth __ $\binom{l}{k}$
multivector extensor field on $U$ _can be identified_ to a $\binom{l}{k}$
multivector extensor over $\mathcal{V}(U).$ It is also true that a smooth __
$\binom{l}{k}$ multiform extensor field on $U$ _can be properly seen_ as a
$\binom{l}{k}$ multiform extensor over $\mathcal{V}(U).$

Thus, the set of smooth __ $\binom{l}{k}$ multivector extensor fields on $U$
is just a module over $\mathcal{S}(U)$ which could be denoted by
$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U).$ And, the
set of smooth __ $\binom{l}{k}$ multiform extensor fields on $U$ is also a
module over $\mathcal{S}(U)$ which can be symbolized as
$\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U).$

####  3.6.1 Algebras of Extensor Fields

We define now the exterior products of smooth multivector extensor fields on
$U$ and smooth multiform extensor fields on $U$. We also present the
definitions of smooth multivector extensor fields on $U$ with smooth multiform
extensor fields on $U.$

The exterior product of either multivector extensor fields or multiform
extensor fields $\tau$ and $\sigma$ is defined as

$\left(\tau\wedge\sigma\right)_{(p)}=\tau_{(p)}\wedge\sigma_{(p)}$ (93)

for every $p\in U$.

Each module over $\mathcal{S}(U)$ of either the smooth multivector extensor
fields on $U$ or the smooth multiform extensor fields on $U$ endowed with the
respective exterior product is an associative algebra.

The duality scalar product of a multiform extensor field $\tau$ with a
multivector extensor field $\sigma$ is a scalar extensor field
$\left\langle\tau,\sigma\right\rangle$ defined by

$\left\langle\tau,\sigma\right\rangle_{(p)}=\left\langle\tau_{(p)},\sigma_{(p)}\right\rangle,$
(94)

for every $p\in U.$

The duality left contracted product of a multiform extensor field $\tau$ with
a multivector extensor field $\sigma$ (or, a multivector extensor field $\tau$
with a multiform extensor field $\sigma$) is the multivector extensor field
(respectively, the multiform extensor field) denoted by
$\left\langle\tau,\sigma\right|$ and defined by

$\left\langle\tau,\sigma\right|_{(p)}=\left\langle\tau_{(p)},\sigma_{(p)}\right|,$
(95)

for every $p\in U.$

The duality right contracted product of a multiform extensor field $\tau$ with
a multivector extensor field $\sigma$ (or, a multivector extensor field $\tau$
with a multiform extensor field $\sigma$) is the multiform extensor field
(respectively, the multivector extensor field) named as
$\left|\tau,\sigma\right\rangle$, and defined by

$\left|\tau,\sigma\right\rangle_{(p)}=\left|\tau_{(p)},\sigma_{(p)}\right\rangle$
(96)

for every $p\in U.$

As in the case of multivectors and multiforms, each duality contracted product
of smooth multivector extensor fields on $U$ with smooth multiform fields on
$U$ yields a non-associative algebra.

##  4 Parallelism Structure and Covariant Derivatives

In this section we present a theory of a general parallelism structure on an
arbitrary real differentiable manifold $M$ of dimension $n$. We present, with
this structure, a detailed theory of the covariant derivatives, deformed
covariant derivatives and relative covariant derivatives of multivector,
multiform and extensor fields.

We detailed some particular formulas valid for a symmetric parallelism
structure. These concepts play an important role in different physical
theories, in particular they are essential for those that want to have a deep
understanding of the geometric theories of the gravitational field.(see, e.g.,
[12, 29, 3]) and do not want to mislead the curvature of a connection defined
in a manifold $M$ ($dimM=m)$ with the fact that $M$ may be a bended
submanifold (a brane) embedded in a Euclidean or speudo-Euclidean manifold
$\mathring{M}\simeq\mathbb{R}^{n}$ ($n>m$). Bending of a brane is
characterized by the shape extensor. See [28] for details.

###  4.1 Parallelism Structure

Let $U$ be an open set of the smooth manifolf $M.$

###### Definition 24

A connection for $M$ is a smooth $2$ vector variables vector operator field
$U\subset M$,

$\Gamma:\mathcal{V}(U)\times\mathcal{V}(U)\longrightarrow\mathcal{V}(U),$

for all $U\subset M$ such that it satisfies the following axioms:

_(i)_ for all $f,g\in\mathcal{S}(U)$ and $a,b,v\in\mathcal{V}(U)$

$\Gamma(fa+gb,v)=f\Gamma(a,v)+g\Gamma(b,v),$ (97)

_(ii_) for all $f,g\in\mathcal{S}(U)$ and $a,v,w\in\mathcal{V}(U)$

$\Gamma(a,fv+gw)=(af)v+(ag)w+f\Gamma(a,v)+g\Gamma(a,w),$ (98)

The behavior of $\Gamma$ with respect to the first variable will be called
_strong linearity_ , and the behavior of $\Gamma$ with respect to the second
variable will be called _quasi linearity_.

The restriction of $\Gamma$ to $U,$ may be called _parallelism structure_ on
$U.$ We will denote this by $\left\langle U,\Gamma\right\rangle$.

###  4.2 Covariant Derivative of Multivector and Multiform Fields

###### Definition 25

The a _-Directional Covariant Derivative_ (a _-DCD_)__ of a smooth multivector
field on $U$, associated with $\left\langle U,\Gamma\right\rangle$, is the
mapping

${\displaystyle\bigwedge}\mathcal{V}(U)\ni
X\mapsto\nabla_{a}X\in{\displaystyle\bigwedge}\mathcal{V}(U),$

such that the following axioms are satisfied:

_(i)_ For all $f\in\mathcal{S}(U):$

$\nabla_{a}f=af.$ (99)

_(ii)_ For all $X^{k}\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U)$ with
$k\geq 1:$

$\displaystyle\nabla_{a}X^{k}(\omega^{1},\ldots,\omega^{k})$
$\displaystyle=aX^{k}(\omega^{1},\ldots,\omega^{k})$
$\displaystyle-X^{k}(\nabla_{a}\omega^{1},\ldots,\omega^{k})\cdots-X^{k}(\omega^{1},\ldots,\nabla_{a}\omega^{k}),$
(100)

for every $\omega^{1},\ldots,\omega^{k}\in\mathcal{V}^{\star}(U).$

_(iii)_ For all $X\in{\textstyle\bigwedge}\mathcal{V}(U)$, if
$X=\overset{n}{\underset{k=0}{{\textstyle\sum}}}X^{k}$, then

$\nabla_{a}X=\overset{n}{\underset{k=0}{{\textstyle\sum}}}\nabla_{a}X^{k}.$
(101)

The basic properties of the $a$-DCD of smooth multivector fields are:

The $a$-Directional Covariant Derivative Operator ($a$-DCDO) $\nabla_{a}$ when
acting on multivector fields is grade-preserving, i.e.,

$\text{if }X\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U),\text{ then
}\nabla_{a}X\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U).$ (102)

For all $f\in\mathcal{S}(U),$ $a,b\in\mathcal{V}(U)$ and
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$

$\displaystyle\nabla_{a+b}X$ $\displaystyle=\nabla_{a}X+\nabla_{b}X,$
$\displaystyle\nabla_{fa}X$ $\displaystyle=f\nabla_{a}X.$ (103)

For all $f\in\mathcal{S}(U),$ $a\in\mathcal{V}(U)$ and
$X,Y\in{\displaystyle\bigwedge}\mathcal{V}(U)$

$\displaystyle\nabla_{a}(X+Y)$ $\displaystyle=\nabla_{a}X+\nabla_{a}Y,$
$\displaystyle\nabla_{a}(fX)$ $\displaystyle=(af)X+f\nabla_{a}X.$ (104)

For all $a\in\mathcal{V}(U)$ and
$X,Y\in{\displaystyle\bigwedge}\mathcal{V}(U)$

$\nabla_{a}(X\wedge Y)=(\nabla_{a}X)\wedge Y+X\wedge\nabla_{a}Y.$ (105)

###### Definition 26

The $a$-DCD of a smooth multiform field on $U$ associated with $\left\langle
U,\Gamma\right\rangle$ is the mapping

${\displaystyle\bigwedge}\mathcal{V}^{\star}(U)\ni\Phi\mapsto\nabla_{a}\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U),$

such that the following axioms are satisfied:

_(i)_ For all $f\in\mathcal{S}(U):$

$\nabla_{a}f=af.$ (106)

_(ii)_ For all $\Phi_{k}\in{\displaystyle\bigwedge^{k}}\mathcal{V}^{\star}(U)$
with $k\geq 1:$

$\displaystyle\nabla_{a}\Phi_{k}(v_{1},\ldots,v_{k})$
$\displaystyle=a\Phi_{k}(v_{1},\ldots,v_{k})$
$\displaystyle-\Phi_{k}(\nabla_{a}v_{1},\ldots,v_{k})\cdots-\Phi_{k}(v_{1},\ldots,\nabla_{a}v_{k}),$
(107)

for every $v_{1},\ldots,v_{k}\in\mathcal{V}(U).$

_(iii)_ For all $\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U):$ if
$\Phi=\underset{k=0}{\overset{n}{{\textstyle\sum}}}\Phi_{k},$ then

$\nabla_{a}\Phi=\underset{k=0}{\overset{n}{{\textstyle\sum}}}\nabla_{a}\Phi_{k}.$
(108)

The basic properties for the $a$-DCD of smooth multiform fields are:

The $a$-DCDO $\nabla_{a}$ when acting on multiform fields is grade-preserving,
i.e.,

$\text{if
}\Phi\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}^{\star}(U),\text{ then
}\nabla_{a}\Phi\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}^{\star}(U).$
(109)

For all $f\in\mathcal{S}(U),$ $a,b\in\mathcal{V}(U)$ and
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$

$\displaystyle\nabla_{a+b}\Phi$ $\displaystyle=\nabla_{a}\Phi+\nabla_{b}\Phi,$
$\displaystyle\nabla_{fa}\Phi$ $\displaystyle=f\nabla_{a}\Phi.$ (110)

For all $f\in\mathcal{S}(u),$ $a\in\mathcal{V}(U)$ and
$\Phi,\Psi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$

$\displaystyle\nabla_{a}(\Phi+\Psi)$
$\displaystyle=\nabla_{a}\Phi+\nabla_{a}\Psi,$
$\displaystyle\nabla_{a}(f\Phi)$ $\displaystyle=(af)\Phi+f\nabla_{a}\Phi.$
(111)

For all $a\in\mathcal{V}(U)$ and
$\Phi,\Psi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$

$\nabla_{a}(\Phi\wedge\Psi)=(\nabla_{a}\Phi)\wedge\Psi+\Phi\wedge\nabla_{a}\Psi.$
(112)

We now present three remarkable properties involving the action of
$\nabla_{a}$ on the duality products of multivector and multiform fields.

###### Proposition 27

When $\nabla_{a}$ acts on the duality scalar product of
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$ with
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$ follows the Leibniz rule, i.e.,

$a\left\langle\Phi,X\right\rangle=\left\langle\nabla_{a}\Phi,X\right\rangle+\left\langle\Phi,\nabla_{a}X\right\rangle.$
(113)

$\nabla_{a}$ acting on the duality left contracted product of
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$ with
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$ (or,
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$ with
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$) satisfies the Leibniz
rule, i.e.,

$\displaystyle\nabla_{a}\left\langle\Phi,X\right|$
$\displaystyle=\left\langle\nabla_{a}\Phi,X\right|+\left\langle\Phi,\nabla_{a}X\right|,$
(114) $\displaystyle\nabla_{a}\left\langle X,\Phi\right|$
$\displaystyle=\left\langle\nabla_{a}X,\Phi\right|+\left\langle
X,\nabla_{a}\Phi\right|.$ (115)

$\nabla_{a}$ acting on the duality right contracted product of
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$ with
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$ (or,
$X\in{\displaystyle\bigwedge}\mathcal{V}^{{}^{\star}}(U)$ with
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}(U)$) satisfies the Leibniz rule,
i.e.,

$\displaystyle\nabla_{a}\left|\Phi,X\right\rangle$
$\displaystyle=\left|\nabla_{a}\Phi,X\right\rangle+\left|\Phi,\nabla_{a}X\right\rangle,$
(116) $\displaystyle\nabla_{a}\left|X,\Phi\right\rangle$
$\displaystyle=\left|\nabla_{a}X,\Phi\right\rangle+\left|X,\nabla_{a}\Phi\right\rangle.$
(117)

Proof. We will prove only the statement given by Eq.(114). Take
$\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U),$
$X\in{\displaystyle\bigwedge}\mathcal{V}(U)$ and
$\Psi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$. By a property of the
duality left contracted product, we have

$\left\langle\left\langle\Phi,X\right|,\Psi\right\rangle=\left\langle
X,\widetilde{\Phi}\wedge\Psi\right\rangle.$ (a)

Now, by using Eq.(113) and Eq.(112), we can write

$\displaystyle\left\langle\nabla_{a}\left\langle\Phi,X\right|,\Psi\right\rangle+\left\langle\left\langle\Phi,X\right|,\nabla_{a}\Psi\right\rangle$
$\displaystyle=\left\langle\nabla_{a}X,\widetilde{\Phi}\wedge\Psi\right\rangle+\left\langle
X,\nabla_{a}(\widetilde{\Phi}\wedge\Psi)\right\rangle$
$\displaystyle=\left\langle\nabla_{a}X,\widetilde{\Phi}\wedge\Psi\right\rangle+\left\langle
X,(\nabla_{a}\widetilde{\Phi})\wedge\Psi)\right\rangle+\left\langle
X,\widetilde{\Phi}\wedge\nabla_{a}\Psi)\right\rangle.$ (b)

Thus, taking into account Eq.(109) and by recalling once again a property of
the duality left contracted product, it follows that

$\left\langle\nabla_{a}\left\langle\Phi,X\right|,\Psi\right\rangle=\left\langle\left\langle\Phi,\nabla_{a}X\right|,\Psi\right\rangle+\left\langle\left\langle\nabla_{a}\Phi,X\right|,\Psi)\right\rangle.$
(c)

Then, by the non-degeneracy of the duality scalar product, the required result
follows.

###  4.3 Covariant Derivative of Extensor Fields

###### Definition 28

Let $\left\langle U,\Gamma\right\rangle$ be a parallelism structure on $U,$
and let us take $a\in\mathcal{V}(U)$. The $a$_-DCD,_ associated with
$\left\langle U,\Gamma\right\rangle$,__ of a smooth multivector extensor field
on $U$ or a smooth multiform extensor field on $U$ are the mappings

$\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)\ni\tau\mapsto\nabla_{a}\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U),$

and

$\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)\ni\tau\mapsto\nabla_{a}\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U),$

such that for all
$X_{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}(U),\ldots,X_{k}\in{\displaystyle\bigwedge\nolimits_{k}^{\Diamond}}\mathcal{V}(U),$
and for all
$\Phi^{1}\in{\displaystyle\bigwedge\nolimits_{1}^{\Diamond}}\mathcal{V}^{\ast}(U),\ldots,$
$\Phi^{l}\in{\displaystyle\bigwedge\nolimits_{l}^{\Diamond}}\mathcal{V}^{\ast}(U)$
we have

$\displaystyle\nabla_{a}\tau(X_{1},...,X_{k},\Phi^{1},...,\Phi^{l})$
$\displaystyle=\nabla_{a}(\tau(X_{1},...,X_{k},\Phi^{1},...,\Phi^{l}))$
$\displaystyle-\tau(\nabla_{a}X_{1},...,X_{k},\Phi^{1},...,\Phi^{l})-\cdots$
$\displaystyle-\tau(X_{1},...,\nabla_{a}X_{k},\Phi^{1},...,\Phi^{l})$
$\displaystyle-\tau(X_{1},...,X_{k},\nabla_{a}\Phi^{1},...,\Phi^{l})-\cdots$
$\displaystyle-\tau(X_{1},...,X_{k},\Phi^{1},...,\nabla_{a}\Phi^{l}).$ (118)

The covariant derivative of smooth multivector (or multiform) extensor fields
has two basic properties:

(i) For $f\in\mathcal{S}(U),$ and $a,b\in\mathcal{V}(U),\mathcal{\ }$and
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$ (or
$\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$)

$\displaystyle\nabla_{a+b}\tau$ $\displaystyle=\nabla_{a}\tau+\nabla_{b}\tau$
(119) $\displaystyle\nabla_{fa}\tau$ $\displaystyle=f\nabla_{a}\tau.$ (120)

(ii) For $f\in\mathcal{S}(U),$ and $a\in\mathcal{V}(U),\mathcal{\ }$and
$\tau,\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$
(or $\tau,\sigma\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$)

$\displaystyle\nabla_{a}(\tau+\sigma)$
$\displaystyle=\nabla_{a}\tau+\nabla_{a}\sigma,$ (121)
$\displaystyle\nabla_{a}(f\tau)$ $\displaystyle=(af)\tau+f\nabla_{a}\tau.$
(122)

The covariant differentiation of the exterior product of smooth multivector
(or multiform) extensor fields satisfies the Leibniz’s rule. We have

###### Proposition 29

For all
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}\mathcal{V}(U)$
(or, $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\sigma\in\left.\overset{\ast}{ext}\right._{r}^{s}\mathcal{V}(U)$), it holds

$\nabla_{a}(\tau\wedge\sigma)=(\nabla_{a}\tau)\wedge\sigma+\tau\wedge\left(\nabla_{a}\sigma\right).$
(123)

Proof. Without loss of generality, we prove this statement only for
multivector extensor fields $(X,\Phi)\mapsto\tau(X,\Phi)$ and
$(Y,\Psi)\mapsto\sigma(Y,\Psi).$ Using Eq.(118), we can write

$\displaystyle\nabla_{a}(\tau\wedge\sigma)(X,Y,\Phi,\Psi)$
$\displaystyle=\nabla_{a}((\tau\wedge\sigma)(X,Y,\Phi,\Psi))$
$\displaystyle-(\tau\wedge\sigma)(\nabla_{a}X,Y,\Phi,\Psi)-(\tau\wedge\sigma)(X,\nabla_{a}Y,\Phi,\Psi)$
$\displaystyle-(\tau\wedge\sigma)(X,Y,\nabla_{a}\Phi,\Psi)-(\tau\wedge\sigma)(X,Y,\Phi,\nabla_{a}\Psi).$

Using Eq.(93) and recalling Leibniz’s rule for the covariant differentiation
of the exterior product of multivector fields, we have

$\displaystyle\nabla_{a}(\tau\wedge\sigma)(X,Y,\Phi,\Psi)$
$\displaystyle=\nabla_{a}(\tau(X,\Phi))\wedge\sigma(Y,\Psi)+\tau(X,\Phi)\wedge\nabla_{a}(\sigma(Y,\Psi))$
$\displaystyle-\tau(\nabla_{a}X,\Phi)\wedge\sigma(Y,\Psi)-\tau(X,\Phi)\wedge\sigma(\nabla_{a}Y,\Psi)$
$\displaystyle-\tau(X,\nabla_{a}\Phi)\wedge\sigma(Y,\Psi)-\tau(X,\Phi)\wedge\sigma(Y,\nabla_{a}\Psi),$

i.e.,

$\displaystyle\nabla_{a}(\tau\wedge\sigma)(X,Y,\Phi,\Psi)$
$\displaystyle=(\nabla_{a}(\tau(X,\Phi))-\tau(\nabla_{a}X,\Phi)-\tau(X,\nabla_{a}\Phi))\wedge\sigma(Y,\Psi)$
$\displaystyle+\tau(X,\Phi)\wedge(\nabla_{a}(\sigma(Y,\Psi))-\sigma(\nabla_{a}Y,\Psi)-\sigma(Y,\nabla_{a}\Psi)).$

Then, using once again Eq.(118) and Eq.(93), the expected result follows.

The covariant differentiation of the duality scalar product and each one of
the duality contracted products of smooth extensor fields satisfies the
Leibniz’s rule.

###### Proposition 30

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}\mathcal{V}(U)$
(or,
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\tau\in\left.\overset{\ast}{ext}\right._{r}^{s}\mathcal{V}(U)$), we have that

$\nabla_{a}\left\langle\tau,\sigma\right\rangle=\left\langle\nabla_{a}\tau,\sigma\right\rangle+\left\langle\tau,\nabla_{a}\sigma\right\rangle.$
(124)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}\mathcal{V}(U)$
(or,
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\tau\in\left.\overset{\ast}{ext}\right._{r}^{s}\mathcal{V}(U)$), it holds

$\displaystyle\nabla_{a}\left\langle\tau,\sigma\right|$
$\displaystyle=\left\langle\nabla_{a}\tau,\sigma\right|+\left\langle\tau,\nabla_{a}\sigma\right|,$
(125) $\displaystyle\nabla_{a}\left\langle\sigma,\tau\right|$
$\displaystyle=\left\langle\nabla_{a}\sigma,\tau\right|+\left\langle\sigma,\nabla_{a}\tau\right|.$
(126)

For all $\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\sigma\in\left.\overset{\left.{}\right.}{ext}\right._{r}^{s}\mathcal{V}(U)$
(or,
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U)$ and
$\tau\in\left.\overset{\ast}{ext}\right._{r}^{s}\mathcal{V}(U)$), it holds

$\displaystyle\nabla_{a}\left|\tau,\sigma\right\rangle$
$\displaystyle=\left|\nabla_{a}\tau,\sigma\right\rangle+\left|\tau,\nabla_{a}\sigma\right\rangle,$
(127) $\displaystyle\nabla_{a}\left|\tau,\sigma\right\rangle$
$\displaystyle=\left|\nabla_{a}\tau,\sigma\right\rangle+\left|\tau,\nabla_{a}\sigma\right\rangle.$
(128)

Proof. We present only the proof of the property given by Eq.(125). Without
loss of generality, we prove this statement only for a multiform extensor
field $\tau$ and a multivector extensor field $\sigma$ such that
$(X,\Phi)\mapsto\tau(X,\Phi)$ and $(Y,\Psi)\mapsto\sigma(Y,\Psi)$.

$\begin{array}[c]{ll}\nabla_{a}\left\langle\tau,\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.&=\nabla_{a}\left(\left\langle\tau,\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.\right)-\left\langle\tau,\sigma\left|\left(\nabla_{a}X,Y,\Phi,\Psi\right)\right.\right.\\\
&-\left\langle\tau,\sigma\left|\left(X,\nabla_{a}Y,\Phi,\Psi\right)\right.\right.-\left\langle\tau,\sigma\left|\left(X,Y,\nabla_{a}\Phi,\Psi\right)\right.\right.\\\
&-\left\langle\tau,\sigma\left|\left(X,Y,\Phi,\nabla_{a}\Psi\right)\right.\right.\end{array}$

or recalling that
$\left\langle\tau,\sigma\left|{}_{\left(p\right)}\right.\right.=\left\langle\tau_{\left(p\right)},\sigma_{\left(p\right)}\left|{}\right.\right.$,

$\begin{array}[c]{ll}\nabla_{a}\left\langle\tau,\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.&=\nabla_{a}\left(\left\langle\tau\left(X,\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.\right)-\left\langle\tau\left(\nabla_{a}X,\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.\\\
&-\left\langle\tau\left(X,\Phi\right),\sigma\left(\nabla_{a}Y,\Psi\right)\left|{}\right.\right.-\left\langle\tau\left(X,\nabla_{a}\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.\\\
&-\left\langle\tau\left(X,\Phi\right),\sigma\left(Y,\nabla_{a}\Psi\right)\left|{}\right.\right..\end{array}$
(129)

On the other hand, from Eq.(116), we can write

$\nabla_{a}\left(\left\langle\tau\left(X,\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.\right)=\left\langle\nabla_{a}\tau\left(X,\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.+\left\langle\tau\left(X,\Phi\right),\nabla_{a}\sigma\left(Y,\Psi\right)\left|{}\right.\right.,$

Eq.(129) can by written as

$\begin{array}[c]{l}\nabla_{a}\left\langle\tau,\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.\\\
=\left\langle\nabla_{a}\tau\left(X,\Phi\right)-\tau\left(\nabla_{a}X,\Phi\right)-\tau\left(X,\nabla_{a}\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.\\\
+\left\langle\tau\left(X,\Phi\right),\nabla_{a}\sigma\left(Y,\Psi\right)-\sigma\left(\nabla_{a}Y,\Psi\right)-\sigma\left(Y,\nabla_{a}\Psi\right)\left|{}\right.\right.\\\
=\left\langle\left(\nabla_{a}\tau\right)\left(X,\Phi\right),\sigma\left(Y,\Psi\right)\left|{}\right.\right.+\left\langle\tau\left(X,\Phi\right),\left(\nabla_{a}\sigma\right)\left(Y,\Psi\right)\left|{}\right.\right.\\\
=\left\langle\nabla_{a}\tau,\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.+\left\langle\tau,\nabla_{a}\sigma\left|\left(X,Y,\Phi,\Psi\right)\right.\right.,\end{array}$

which proves our result.

Finally we prove that the duality adjoint operator commutes with the $a$-DCDO,
i.e., we have

###### Proposition 31

If $\tau$ is any one of the four smooth one-variable extensor fields on $U$,
then

$(\nabla_{a}\tau)^{\bigtriangleup}=\nabla_{a}\tau^{\bigtriangleup}.$ (130)

Proof. Without loss of generality, we prove this statement only for $\tau\in
ext({\displaystyle\bigwedge\nolimits_{1}^{\diamond}}\mathcal{V}(U),{\displaystyle\bigwedge\nolimits^{\diamond}}\mathcal{V}(U)).$

Let us $X\in{\displaystyle\bigwedge\nolimits_{1}^{\diamond}}\mathcal{V}(U)$
and
$\Phi\in{\displaystyle\bigwedge\nolimits^{\diamond}}\mathcal{V}^{\ast}(U).$ We
must prove that

$\left\langle\nabla_{a}\tau^{\bigtriangleup}(\Phi),X\right\rangle=\left\langle\Phi,\nabla_{a}\tau(X)\right\rangle.$

By using Eq.(118) and recalling the Leibniz’s rule for the covariant
differentiation of the duality scalar product of multiform fields with
multivector fields, we can write

$\left\langle\nabla_{a}\tau^{\bigtriangleup}(\Phi),X\right\rangle=\left\langle\nabla_{a}(\tau^{\bigtriangleup}(\Phi)),X\right\rangle-\left\langle\tau^{\bigtriangleup}(\nabla_{a}(\Phi)),X\right\rangle,$

and from Eq.(113)

$\left\langle\nabla_{a}\tau^{\bigtriangleup}(\Phi),X\right\rangle=a\left\langle\tau^{\bigtriangleup}(\Phi),X\right\rangle-\left\langle\tau^{\bigtriangleup}(\Phi),\nabla_{a}X\right\rangle-\left\langle\tau^{\bigtriangleup}(\nabla_{a}\Phi),X\right\rangle.$

Recalling now the fundamental property of the duality adjoint, we get

$\left\langle\nabla_{a}\tau^{\bigtriangleup}(\Phi),X\right\rangle=a\left\langle\Phi,\tau(X)\right\rangle-\left\langle\Phi,\tau(\nabla_{a}X)\right\rangle-\left\langle\nabla_{a}\Phi,\tau(X)\right\rangle.$

Using once again the Leibniz’s rule, Eq.(113), for the covariant
differentiation of the duality scalar product we get

$\left\langle\nabla_{a}\tau^{\bigtriangleup}(\Phi),X\right\rangle=\left\langle\Phi,\nabla_{a}(\tau(X))\right\rangle-\left\langle\Phi,\tau(\nabla_{a}X)\right\rangle,$

and thus, using once again Eq.(118), the required result follows.

In particular, the $a$_-DCD,_ _of a smooth_ $k$_-covariant and_
$l$_-contravariant vector_(_or form_)_extensor field_ on $U,$ and associated
with $\left\langle U,\Gamma\right\rangle,$ are defined by

$\left.\overset{}{ext}\right._{k}^{l}\mathcal{V}(U)\ni\tau\mapsto\nabla_{a}\tau\in\left.\overset{}{ext}\right._{k}^{l}\mathcal{V}(U)\text{
and
}\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U)\ni\tau\mapsto\nabla_{a}\tau\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U),$

respectively, such that for every $v_{1},\ldots,v_{k}\in\mathcal{V}(U),$ and
$\omega^{1},\ldots,\omega^{l}\in\mathcal{V}^{\ast}(U)$:

$\displaystyle\nabla_{a}\tau(v_{1},\ldots,v_{k},\omega^{1},\ldots,\omega^{l})$
$\displaystyle=\nabla_{a}(\tau(v_{1},\ldots,v_{k},\omega^{1},\ldots,\omega^{l}))$
$\displaystyle-\tau(\nabla_{a}v_{1},\ldots,v_{k},\omega^{1},\ldots,\omega^{l})-\cdots$
$\displaystyle-\tau(v_{1},\ldots,\nabla_{a}v_{k},\omega^{1},\ldots,\omega^{l})$
$\displaystyle-\tau(v_{1},\ldots,v_{k},\nabla_{a}\omega^{1},\ldots,\omega^{l})-\cdots$
$\displaystyle-\tau(v_{1},\ldots,v_{k},\omega^{1},\ldots,\nabla_{a}\omega^{l}).$
(131)

Where we can easily see that it meets the basic properties (119), (120), (121)
and (122).

It is also worth recalling here that the $a$_-DCD_ of a smooth vector field on
$U$, associated with $\left\langle U,\Gamma\right\rangle$, is the mapping

$\mathcal{V}(U)\ni v\mapsto\nabla_{a}v\in\mathcal{V}(U)$

such that

$\nabla_{a}v=\Gamma(a,v),$ (132)

and the $a$_-DCD_ of a smooth form field on $U,$ is the mapping

$V^{\ast}(U)\ni\omega\mapsto\nabla_{a}\omega\in\mathcal{V}^{\ast}(U)$

such that for every $v\in\mathcal{V}(U)$

$\nabla_{a}\omega(v)=a\omega(v)-\omega(\nabla_{a}v).$ (133)

###  4.4 Deformed Covariant Derivative

Let $\left\langle U,\Gamma\right\rangle$ be a parallelism structure on $U.$
Let us take an invertible smooth extensor operator field $\lambda$ on
$V\supseteq U,$ i.e., $\lambda:\mathcal{V}(U)\rightarrow\mathcal{V}(U).$ The
$\lambda$_-deformation_ of $\left\langle U,\Gamma\right\rangle$ is a well-
defined connection on $U,$ namely $\overset{\lambda}{\Gamma}$, given by

$\mathcal{V}(U)\times\mathcal{V}(U)\ni(a,v)\mapsto\overset{\lambda}{\Gamma}(a,v)\in\mathcal{V}(U)$

such that

$\overset{\lambda}{\Gamma}(a,v)=\lambda(\Gamma(a,\lambda^{-1}(v))).$ (134)

It is easy to see that $\overset{\lambda}{\Gamma}$ is indeed a connection on
$U$ since it satisfies Eq.(97) and Eq.(98).

The parallelism structure $\left\langle
U,\overset{\lambda}{\Gamma}\right\rangle$ is said to be the
$\lambda$_-deformation_ of $\left\langle U,\Gamma\right\rangle$.

Let us take $a\in\mathcal{V}(U).$ The $a$-_DCDO_ associated with $\left\langle
U,\overset{\lambda}{\Gamma}\right\rangle,$ namely
$\overset{\lambda}{\nabla}_{a}$, has the basic properties:

###### Proposition 32

For all $v\in\mathcal{V}(U)$

$\overset{\lambda}{\nabla}_{a}v=\lambda(\nabla_{a}\lambda^{-1}(v)).$ (135)

It follows from Eq.(132) and Eq.(134).

For all $\omega\in\mathcal{V}^{\ast}(U)$

$\overset{\lambda}{\nabla}_{a}\omega=\lambda^{-\bigtriangleup}(\nabla_{a}\lambda^{\bigtriangleup}(\omega)).$
(136)

Proof. Eq.(135 follows from Eq.(132) and Eq.(134)). To prove Eq.(136) take
$v\in\mathcal{V}(U)$. Then using Eq.(133) and Eq.(135), we have

$\displaystyle\overset{\lambda}{\nabla}_{a}\omega(v)$
$\displaystyle=a\omega(v)-\omega(\overset{\lambda}{\nabla}_{a}v)$
$\displaystyle=a\omega(v)-\omega(\lambda(\nabla_{a}\lambda^{-1}(v)))$
$\displaystyle=a\left\langle\omega,v\right\rangle-\left\langle\omega,\lambda(\nabla_{a}\lambda^{-1}(v))\right\rangle,$
(137)

but, by recalling the fundamental property of the _duality adjoint_ and by
using once again Eq.(133), the second term in Eq.(137) can be written

$\displaystyle\left\langle\omega,\lambda(\nabla_{a}\lambda^{-1}(v))\right\rangle$
$\displaystyle=\left\langle\lambda^{\bigtriangleup}(\omega),\nabla_{a}\lambda^{-1}(v)\right\rangle$
$\displaystyle=a\left\langle\lambda^{\bigtriangleup}(\omega),\lambda^{-1}(v)\right\rangle-\left\langle\nabla_{a}\lambda^{\bigtriangleup}(\omega),\lambda^{-1}(v)\right\rangle$
$\displaystyle=a\left\langle\omega,v\right\rangle-\left\langle\lambda^{-\bigtriangleup}\nabla_{a}\lambda^{\bigtriangleup}(\omega),v\right\rangle,$
(138)

Finally, putting Eq.(138) into Eq.(137), the expected result follows.

####  4.4.1 Deformed Covariant Derivative of Multivector and Multiform Fields

Recall that $\lambda^{\bigtriangleup}$ is the so-called _duality adjoint_ of
$\lambda,$ and $\lambda^{-\bigtriangleup}$ is a _short notation_ for
$(\lambda^{\bigtriangleup})^{-1}=(\lambda^{-1})^{\bigtriangleup}$. We give now
two properties of $\overset{\lambda}{\text{ }\nabla}_{a}$ which are
generalizations of the basic properties (135) and (136) above.

###### Proposition 33

For all $X\in{\displaystyle\bigwedge}\mathcal{V}(U)$

$\overset{\lambda}{\nabla}_{a}X=\underline{\lambda}(\nabla_{a}\underline{\lambda}^{-1}(X)),$
(139)

where $\underline{\lambda}$ is the so-called _exterior power extension_ of
$\lambda,$ and $\underline{\lambda}^{-1}$ is a _more simple notation_ for
$(\underline{\lambda})^{-1}=\underline{(\lambda^{-1})}.$

For all $\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U)$

$\overset{\lambda}{\nabla}_{a}\Phi=\underline{\lambda}^{-\bigtriangleup}(\nabla_{a}\underline{\lambda}^{\bigtriangleup}(\Phi)),$
(140)

where
$\underline{\lambda}^{\bigtriangleup}=(\underline{\lambda})^{\bigtriangleup}=\underline{(\lambda^{\bigtriangleup})}$
and
$\underline{\lambda}^{-\bigtriangleup}=(\underline{\lambda}^{\bigtriangleup})^{-1}.$

Proof. To prove the property for smooth multivector fields as given by
Eq.(139), we make use the following criterion:

(i) We check the statement for scalar fields $f\in\mathcal{S}(U)$ (use
Eq.(99)).

(ii) We check the statement for simple $k$-vector fields
$v_{1}\wedge\cdots\wedge
v_{k}\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U)$ (use mathematical
induction).

(iii) We check the statement for a finite addition of simple $k$-vector fields
$X^{k}+\cdots Z^{k}\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U)$ (use
Eq.(104) and recalling the linear operator character for the extended of a
linear operator)

(iv) Then it becomes easy to prove the statement for multivector fields
$X\in{\textstyle\bigwedge}\mathcal{V}(U)$.

####  4.4.2 Deformed Covariant Derivative of Extensor Fields

We present now two properties for $\overset{\lambda}{\nabla}_{a}$ which are
generalizations of the properties (139) and (140).

###### Proposition 34

For all
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U):$

$\overset{\lambda}{\nabla}_{a}\tau=\underline{\lambda}\nabla_{a}\underline{\lambda}^{-1}\tau,$
(141)

where $\underline{\lambda}^{-1}\tau$ means the action of
$\underline{\lambda}^{-1}$ on the smooth multivector extensor field $\tau,$
and $\underline{\lambda}\nabla_{a}\underline{\lambda}^{-1}\tau$ is the action
of $\underline{\lambda}$ on the smooth multivector extensor field
$\nabla_{a}\underline{\lambda}^{-1}\tau$.

###### Proposition 35

For all $\upsilon\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U):$

$\overset{\lambda}{\nabla}_{a}\upsilon=\underline{\lambda}^{-\bigtriangleup}\nabla_{a}\underline{\lambda}^{\bigtriangleup}\upsilon,$
(142)

where $\underline{\lambda}^{\bigtriangleup}\upsilon$ means the action of
$\underline{\lambda}^{\bigtriangleup}$ on the smooth multiform extensor field
$\upsilon,$ and
$\underline{\lambda}^{-\bigtriangleup}\nabla_{a}\underline{\lambda}^{\bigtriangleup}\upsilon$
is the action of $\underline{\lambda}^{-\bigtriangleup}$ on the smooth
multiform extensor field
$\nabla_{a}\underline{\lambda}^{\bigtriangleup}\upsilon.$

Proof. We will prove only the property for smooth multivector extensor fields
as given by Eq.(141). Without restrictions on generality, we can work with a
multivector extensor field $(X,\Phi)\mapsto\tau(X,\Phi)$.

Then, using Eq.(118) and taking into account the properties (139) and (140),
we can write

$\displaystyle(\overset{\lambda}{\nabla}_{a}\tau)(X,\Phi)$
$\displaystyle=\overset{\lambda}{\nabla}_{a}\tau(X,\Phi)-\tau(\overset{\lambda}{\nabla}_{a}X,\Phi)-\tau(X,\overset{\lambda}{\nabla}_{a}\Phi)$
$\displaystyle=\underline{\lambda}(\nabla_{a}\underline{\lambda}^{-1}\circ\tau(X,\Phi))-\tau(\underline{\lambda}(\nabla_{a}\underline{\lambda}^{-1}(X)),\Phi)-\tau(X,\underline{\lambda}^{-\bigtriangleup}(\nabla_{a}\underline{\lambda}^{\bigtriangleup}(\Phi))),$

i.e.,

$\displaystyle(\underline{\lambda}^{-1}\circ\overset{\lambda}{\nabla}_{a}\tau)(X,\Phi)$
$\displaystyle=\nabla_{a}\underline{\lambda}^{-1}\circ\tau(X,\Phi)-\underline{\lambda}^{-1}\circ\tau(\underline{\lambda}(\nabla_{a}\underline{\lambda}^{-1}(X)),\Phi)$
$\displaystyle-\underline{\lambda}^{-1}\circ\tau(X,\underline{\lambda}^{-\bigtriangleup}(\nabla_{a}\underline{\lambda}^{\bigtriangleup}(\Phi))).$

By recalling the action of an extended operator $\underline{\lambda}^{-1}$ on
a multivector extensor $\tau,$ we get

$\displaystyle(\underline{\lambda}^{-1}\circ\overset{\lambda}{\nabla}_{a}\tau)(X,\Phi)$
$\displaystyle=\nabla_{a}\underline{\lambda}^{-1}\tau(\underline{\lambda}^{-1}(X),\underline{\lambda}^{\bigtriangleup}(\Phi))-\underline{\lambda}^{-1}\tau(\nabla_{a}\underline{\lambda}^{-1}(X)),\underline{\lambda}^{\bigtriangleup}(\Phi))$
$\displaystyle-\underline{\lambda}^{-1}\tau(\underline{\lambda}^{-1}(X)),\nabla_{a}\underline{\lambda}^{\bigtriangleup}(\Phi)).$

Using once again Eq.(118), we have

$(\overset{\lambda}{\nabla}_{a}\tau)(X,\Phi)=\underline{\lambda}\circ(\nabla_{a}\underline{\lambda}^{-1}\tau)(\underline{\lambda}^{-1}(X),\underline{\lambda}^{\bigtriangleup}(\Phi)),$

and finally recalling once again the action of an exterior power extension
operator $\underline{\lambda}$ on a multivector extensor
$\nabla_{a}\underline{\lambda}^{-1}\tau$, the required result follows.

###  4.5 Relative Covariant Derivative

###### Definition 36

Let $\left\\{b_{\mu},\beta^{\mu}\right\\}$ be a pair of dual frame fields for
$U\subseteq M$. Associated with $\left\\{b_{\mu},\beta^{\mu}\right\\}$ we can
construct a well-defined connection on __ $U$ given by the mapping

$B:\mathcal{V}(U)\times\mathcal{V}(U)\longrightarrow\mathcal{V}(U),$

such that

$B(a,v)=\left[a\beta^{\sigma}(v)\right]b_{\sigma}.$ (143)

$B$ is called _relative connection_ on $U$ with respect to __
$\left\\{b_{\mu},\beta^{\mu}\right\\}$ _(_ or simply relative connection for
short _)_.

The parallelism structure $\left\langle U,B\right\rangle$ will be called
_relative parallelism structure_ with respect to
$\left\\{b_{\mu},\beta^{\mu}\right\\}.$

The __ $a$_-DCDO_ induced by the relative connection will be denoted by
$\partial_{a}$. According with Eq.(132), the $a$_-DCD_ of a smooth vector
field is given by

$\partial_{a}v=\left[a\beta^{\sigma}(v)\right]b_{\sigma}.$ (144)

Note that $\partial_{a}$ is the unique $a$_-DCDO_ which satisfies the
condition

$\partial_{a}b_{\mu}=0.$ (145)

On the other hand, given a parallelism structure $\left\langle
U_{0},\Gamma\right\rangle$ and any relative parallelism structure
$\left\langle U,B\right\rangle,$ such that $U_{0}\cap U\neq\phi$. There exists
a smooth $2$_-covariant vector extensor field_ on $U_{0}\cap U$, namely
$\gamma$, defined by

$\mathcal{V}(U_{0}\cap U)\times\mathcal{V}(U_{0}\cap
U)\ni(a,v)\mapsto\gamma(a,v)\in\mathcal{V}(U_{0}\cap U)$

such that

$\gamma(a,v)=\beta^{\mu}(v)\nabla_{a}b_{\mu}$ (146)

which satisfies

$\Gamma(a,v)=B(a,v)+\gamma(a,v).$ (147)

Such a extensor field $\gamma$ will be called the _relative connection
extensor field_ 333The properties of the tensor field
$\gamma_{\mu\nu}^{\alpha}$ such that
$\gamma(\partial_{\mu},\partial_{\nu})=\gamma_{\mu\nu}^{\alpha}\partial_{\alpha}$
where $\\{\partial_{\mu}\\}$ is a basis for $\mathcal{V}(U_{0}\cap U)$ are
studied in details in [27]. on $U_{0}\cap U$.

From Eq.(132),this means that for all $v\in\mathcal{V}(U_{0}\cap U):$

$\nabla_{a}v=\partial_{a}v+\gamma_{a}(v),$ (148)

where $\nabla_{a}$ is the $a$-_DCDO_ associated with $\left\langle
U_{0},\Gamma\right\rangle$ and $\partial_{a}$ is the $a$-_DCDO_ associated
with $\left\langle U,B\right\rangle$ (note that $\gamma_{a}$ is a smooth
_vector operator field_ on $U_{0}\cap U$ defined by
$\gamma_{a}(v)=\gamma(a,v)$).

By using Eq.(133) and Eq.(148), we get that for all
$\omega\in\mathcal{V}^{\ast}(U_{0}\cap U):$

$\nabla_{a}\omega=\partial_{a}\omega-\gamma_{a}^{\bigtriangleup}(\omega),$
(149)

where $\gamma_{a}^{\bigtriangleup}$ is the _dual adjoint_ of $\gamma_{a}$
(i.e.,
$\left\langle\gamma_{a}^{\bigtriangleup}(\omega),v\right\rangle=\left\langle\omega,\gamma_{a}(v)\right\rangle$).

####  4.5.1 Relative Covariant Derivative of Multivector and Multiform Fields

Let $\left\langle U_{0},\Gamma\right\rangle$ be a parallelism structure on
$U_{0},$ and let $\nabla_{a}$ be its associated $a$-DCDO. Take any relative
parallelism structure $\left\langle U,B\right\rangle$ compatible with
$\left\langle U_{0},\Gamma\right\rangle$ (i.e., $U_{0}\cap U\neq\emptyset$).

We present now the split theorem, i.e. a generalization of Eqs. (148) and
(149), for smooth multivector fields and for multiform fields. We know that
for all $v\in\mathcal{V}(U_{0}\cap U)$,
$\nabla_{a}v=\partial_{a}v+\gamma_{a}(v)$, and for all
$\omega\in\mathcal{V}^{\star}(U_{0}\cap U)$,
$\nabla_{a}\omega=\partial_{a}\omega-\gamma_{a}^{\bigtriangleup}(\omega).$

###### Theorem 37

_(a)_ For all $X\in{\textstyle\bigwedge}\mathcal{V}(U_{0}\cap U)$

$\nabla_{a}X=\partial_{a}X+\underset{\smile}{\gamma_{a}}(X),$ (150)

where $\underset{\smile}{\gamma_{a}}$ is _contracted extension operator_
444See appendix 6.3 and [4], to recall the notion of the generalization
procedure._of_ $\gamma_{a}$

_(b)_ For all $\Phi\in{\textstyle\bigwedge}\mathcal{V}^{\star}(U_{0}\cap U)$

$\nabla_{a}\Phi=\partial_{a}\Phi-\underset{\smile}{\gamma_{a}^{\bigtriangleup}}(\Phi),$
(151)

where $\underset{\smile}{\gamma_{a}^{\bigtriangleup}}$ is the so-called
_contracted extension operator_ of $\gamma_{a}^{\bigtriangleup}$ which as we
know coincides with the so-called duality adjoint of
$\underset{\smile}{\gamma_{a}}.$

Proof. To prove the property for smooth multivector fields as given by
Eq.(150), we use as above the following procedure:.

(i) We check the statement for scalar fields $f\in\mathcal{S}(U).$

(ii) Next, we check the statement for simple $k$-vector fields
$v_{1}\wedge\cdots\wedge
v_{k}\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U).$

(iii) Next we check the statement for a finite addition of simple $k$-vector
fields $X^{k}+\cdots
Z^{k}\in{\textstyle\bigwedge\nolimits^{k}}\mathcal{V}(U).$

(iv) We then can easily prove the statement for multivector fields
$X\in{\textstyle\bigwedge}\mathcal{V}(U)$.

Let $\left\langle U,B\right\rangle$ and $\left\langle
U^{\prime},B^{\prime}\right\rangle$, $U\cap U^{\prime}\neq\emptyset$, be two
compatible parallelism structures taken on a smooth manifold $M$. The
$a$-DCDO’s associated with $\left\langle U,B\right\rangle$ and $\left\langle
U^{\prime},B^{\prime}\right\rangle$ are denoted by $\partial_{a}$ and
$\partial_{a}^{\prime}$, respectively. As we already know [5], there exists a
well-defined smooth extensor operator field on $U\cap U^{\prime}$, the
_Jacobian field_ $J$ (see Appendix E), such that the following two basic
properties are satisfied: for all $v\in\mathcal{V}(U\cap U^{\prime})$,
$\partial_{a}^{\prime}v=J(\partial_{a}J^{-1}(v))$, and for all
$\omega\in\mathcal{V}^{\star}(U\cap U^{\prime})$,
$\partial_{a}^{\prime}\omega=J^{-\bigtriangleup}(\partial_{a}J^{\bigtriangleup}(\omega))$.

We can see immediately that the basic properties just recalled implies that
$\partial_{a}^{\prime}$ is the $J$-deformation of $\partial_{a}.$

We present now two properties for the relative covariant derivatives which are
generalizations of the basic properties just recalled above.

###### Proposition 38

For all $X\in{\displaystyle\bigwedge}\mathcal{V}(U\cap U^{\prime})$

$\partial_{a}^{\prime}X=\underline{J}(\partial_{a}\underline{J}^{-1}(X)).$
(152)

For all $\Phi\in{\displaystyle\bigwedge}\mathcal{V}^{\star}(U\cap U^{\prime})$

$\partial_{a}^{\prime}\Phi=\underline{J}^{-\bigtriangleup}(\partial_{a}\underline{J}^{\bigtriangleup}(\Phi)).$
(153)

Proof. Eq.(152) is an immediate consequence of Eq.(139)., Eq.(153) is an
immediate consequence of Eq.(140)_._

####  4.5.2 Relative Covariant Derivative of Extensor Fields

As we know from (150) and (151), there exists a well-defined smooth vector
operator field on $U_{0}\cap U$, called the _relative connection field_
$\gamma_{a}$, which satisfies the _split theorem_ valid for smooth multivector
fields and for smooth multiform fields, i.e.: for all
$X\in{\textstyle\bigwedge}\mathcal{V}(U_{0}\cap
U)$,$\nabla_{a}X=\partial_{a}X+\underset{\smile}{\gamma_{a}}(X)$, and for all
$\Phi\in{\textstyle\bigwedge}\mathcal{V}^{\star}(U_{0}\cap U)$,
$\nabla_{a}\Phi=\partial_{a}\Phi-\underset{\smile}{\gamma_{a}^{\bigtriangleup}}(\Phi)$.

We now present a split theorem for smooth multivector extensor fields and for
smooth multiform extensor fields, which are the generalizations of the
properties just recalled above.

###### Theorem 39

_(i)_ For all
$\tau\in\left.\overset{\left.{}\right.}{ext}\right._{k}^{l}\mathcal{V}(U_{0}\cap
U):$

$\nabla_{a}\tau=\partial_{a}\tau+\underset{\smile}{\gamma_{a}}\tau,$ (154)

where $\underset{\smile}{\gamma_{a}}\tau$ means the action of
$\underset{\smile}{\gamma_{a}}$ on the smooth multivector extensor field
$\tau.$

_(ii)_ For all
$\upsilon\in\left.\overset{\ast}{ext}\right._{k}^{l}\mathcal{V}(U_{0}\cap U):$

$\nabla_{a}\upsilon=\partial_{a}\upsilon-\underset{\smile}{\gamma_{a}^{\bigtriangleup}}\upsilon,$
(155)

where $\underset{\smile}{\gamma_{a}^{\bigtriangleup}}\upsilon$ means the
action of $\underset{\smile}{\gamma_{a}^{\bigtriangleup}}$ on the smooth
multiform extensor field $\upsilon.$

Proof. We prove only the property for smooth multivector extensor fields,
i.e., Eq.(154). Without loss of generality, we check this statement for a
multivector extensor field $(X,\Phi)\mapsto\tau(X,\Phi)$.

Using Eq.(118), and by taking into account the properties just recalled above,
we have

$\displaystyle\nabla_{a}\tau(X,\Phi)$
$\displaystyle=\nabla_{a}\tau(X,\Phi)-\tau(\nabla_{a}X,\Phi)-\tau(X,\nabla_{a}\Phi)$
$\displaystyle=\partial_{a}\tau(X,\Phi)+\underset{\smile}{\gamma_{a}}(\tau(X,\Phi))$
$\displaystyle-\tau(\partial_{a}X+\underset{\smile}{\gamma_{a}}(X),\Phi)-\tau(X,\partial_{a}\Phi-\underset{\smile}{\gamma_{a}^{\bigtriangleup}}(\Phi))$
$\displaystyle=\partial_{a}\tau(X,\Phi)-\tau(\partial_{a}X,\Phi)-\tau(X,\partial_{a}\Phi)$
$\displaystyle+\underset{\smile}{\gamma_{a}}(\tau(X,\Phi))-\tau(\underset{\smile}{\gamma_{a}}(X),\Phi)+\tau(X,\underset{\smile}{\gamma_{a}^{\bigtriangleup}}(\Phi)).$

Then, using once again Eq.(118) and recalling the action of a generalized
operator $\underset{\smile}{\gamma_{a}}$ on a multivector extensor $\tau$, see
Eq.(61), we get

$(\nabla_{a}\tau)(X,\Phi)=(\partial_{a}\tau)(X,\Phi)+(\underset{\smile}{\gamma_{a}}\tau)(X,\Phi),$

and the proposition is proved.

##  5 Torsion and Curvature

###### Definition 40

Let $\left\langle U,\Gamma\right\rangle$ be a parallelism structure on $U$.
The smooth $2$_-covariant vector extensor field_ on $U$, defined by

$\mathcal{V}(U)\times\mathcal{V}(U)\ni(a,b)\mapsto\tau(a,b)\in\mathcal{V}(U)$

such that

$\tau(a,b)=\nabla_{a}b-\nabla_{b}a-\left[a,b\right],$ (156)

will be called the _fundamental torsion extensor field 555$\tau$ is skew-
symmetric, i.e.,  $\tau\left(b,a\right)=-\tau\left(a,b\right).$ _ of
$\left\langle U,\Gamma\right\rangle$.

Accordingly, there exists a smooth $(2,1)$_-extensor field_ on $U,$ defined by

${\textstyle\bigwedge\nolimits^{2}}\mathcal{V}(U)\ni
X^{2}\mapsto\mathcal{T}(X^{2})\in\mathcal{V}(U)$

such that

$\mathcal{T}(X^{2})=\frac{1}{2}\left\langle\varepsilon^{\mu}\wedge\varepsilon^{\nu},X^{2}\right\rangle\tau(e_{\mu},e_{\nu}),$
(157)

where $\left\\{e_{\mu},\varepsilon^{\mu}\right\\}$ is any pair of dual frame
fields on $V\supseteq U.$ It should be emphasized that $\mathcal{T}$, as
extensor field associated with $\tau$, is well-defined since the vector field
$\mathcal{T}(X^{2})$ does not depend on the choice of
$\left\\{e_{\mu},\varepsilon^{\mu}\right\\}.$

It should be remarked that the well known _torsion tensor field,_ see, e.g.,
[1, 27],__ is just given by

$\mathcal{V}(U)\times\mathcal{V}(U)\times\mathcal{V}^{\ast}(U)\ni(a,b,\omega)\mapsto
T(a,b,\omega)\in\mathcal{S}(U)$

such that

$T(a,b,\omega)=\left\langle\omega,\tau(a,b)\right\rangle.$ (158)

Note that it is possible to get $\tau$ in terms of $T,$ i.e.,

$\tau(a,b)=T(a,b,\varepsilon^{\mu})e_{\mu}.$ (159)

It is also possible to introduce a _third torsion extensor field_ for
$\left\langle U,\Gamma\right\rangle$ by defining the smooth $(1,2)$_-extensor
field_ on $U$,

$\mathcal{V}^{\ast}(U)\ni\omega\mapsto\Theta(\omega)\in{\textstyle\bigwedge\nolimits^{2}}\mathcal{V}^{\ast}(U),$

such that

$\Theta(\omega)=\frac{1}{2}\left\langle\omega,\tau(e_{\mu},e_{\nu})\right\rangle\varepsilon^{\mu}\wedge\varepsilon^{\nu},$
(160)

We call $\Theta$ the _Cartan torsion extensor field_ of $\left\langle
U,\Gamma\right\rangle$.

###### Definition 41

The smooth $3$_-covariant extensor vector field_ on $U$, defined by

$\mathcal{V}(U)\times\mathcal{V}(U)\times\mathcal{V}(U)\ni(a,b,c)\mapsto\rho(a,b,c)\in\mathcal{V}(U),$

such that

$\rho(a,b,c)=\left[\nabla_{a},\nabla_{b}\right]c-\nabla_{\left[a,b\right]}c,$
(161)

will be called the _fundamental curvature extensor field 666As can be easily
verified, $\rho$ is skew-symmetric with respect to the first and the second
variables, i.e.,$\rho(b,a,c)=-\rho(a,b,c).$_ of $\left\langle
U,\Gamma\right\rangle$.

Thus, there exists a smooth $1$_bivector and_ $1$_vector variables vector
extensor field_ on $U$, defined by

${\displaystyle\bigwedge\nolimits^{2}}\mathcal{V}(U)\times\mathcal{V}(U)\ni(X^{2},c)\mapsto\mathcal{R}(X^{2},c)\in\mathcal{V}(U)$

such that

$\mathcal{R}(X^{2},c)=\frac{1}{2}\left\langle\varepsilon^{\mu}\wedge\varepsilon^{\nu},X^{2}\right\rangle\rho(e_{\mu},e_{\nu},c),$
(162)

where $\left\\{e_{\mu},\varepsilon^{\mu}\right\\}$ is any pair of dual frame
fields on $V\supseteq U.$ We note that $\mathcal{R},$ as extensor field
associated with $\rho,$ is well-defined since the vector field
$\mathcal{R}(X^{2},c)$ does not depend on the choice of
$\left\\{e_{\mu},\varepsilon^{\mu}\right\\}.$

###### Proposition 42

The fundamental curvature field $\rho$ satisfies a _cyclic property_ , i.e.,

$\rho(a,b,c)+\rho(b,c,a)+\rho(c,a,b)=0.$

Proof. Let us take $a,b,c\in\mathcal{V}(U).$ By using Eq.(161), we can write

$\displaystyle\rho(a,b,c)$
$\displaystyle=\nabla_{a}\nabla_{b}c-\nabla_{b}\nabla_{a}c-\nabla_{\left[a,b\right]}c,$
(a) $\displaystyle\rho(b,c,a)$
$\displaystyle=\nabla_{b}\nabla_{c}a-\nabla_{c}\nabla_{b}a-\nabla_{\left[b,c\right]}a,$
(b) $\displaystyle\rho(c,a,b)$
$\displaystyle=\nabla_{c}\nabla_{a}b-\nabla_{a}\nabla_{c}b-\nabla_{\left[c,a\right]}b.$
(c)

By adding Eqs. (a), (b) and (c), we have

$\displaystyle\rho(a,b,c)+\rho(b,c,a)+\rho(c,a,b)$
$\displaystyle=\nabla_{a}(\nabla_{b}c-\nabla_{c}b)+\nabla_{b}(\nabla_{c}a-\nabla_{a}c)+\nabla_{c}(\nabla_{a}b-\nabla_{b}a)$