# On Connections and their Curvatures

Gustavo Amilcar Saldaña Moncada & Gregor Weingart  Gustavo Amilcar Saldaña
Moncada
Instituto de Matemáticas (Ciudad de México) UNAM<EMAIL_ADDRESS>Gregor Weingart
Instituto de Matemáticas (Cuernavaca) UNAM<EMAIL_ADDRESS>
###### Abstract.

This paper presents a brief study on connections on fiber, principal and
vector smooth bundles as well as some relations with their curvatures.

MSC 2010: 57R22.

Keywords: Fiber bundles, non–linear connections, curvature

##  1\. Introduction

The study of bundles and their ad hoc  connections is one of the must
important subjects in differential geometry. For example, the concept of
vector bundle and linear connection appear on every introductory course in the
form of the tangent bundle and the Levi–Civita connection, respectively.
Principal bundles also play an important role; for example the frame bundle of
a Riemannian manifold reflects different ways to give an orthonormal basis for
tangent spaces, not to mentiontion that field theories in modern physics are
developed by using principal bundles and principal connections [1]. All these
objects arise from the more general concepts of fiber bundles and non–linear
connections. This work’s aim is to present a brief study on connections on
fiber, principal and vector smooth bundles as well as some relations with
their curvatures.

This paper corresponds to an extended version of the second section of [3],
but focused in geometrical relations, not the functioral ones and it breaks
down as follows: in the second section we present the basics of the theory of
fiber bundles and non–linear connections, as well as the relations of the
curvature. The third section is about principal connections on principal
bundles; while the fourth section is about vector bundles and linear
connections. Finally in the appendix A we present some calculations related to
the curvature and its relation with covariant derivatives in local
coordinates.

##  2\. Non–Linear Connection

A fiber bundle over a manifold $M$ with model fiber manifold $\mathscr{F}$ is
a manifold $\mathscr{F}M$ endowed with a smooth projection map
$\pi:\,\mathscr{F}M\longrightarrow M$, which is locally trivializable in the
sense that there exists an open covering of $M$ by open subsets
$U\,\subset\,M$, each of which allows for a diffeomorphism

$\Phi:\;\;U\;\times\;\mathscr{F}\;\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\;\pi^{-1}(\,U\,)$

such that $\pi\,\circ\,\Phi:\,U\times\mathscr{F}\longrightarrow U$ equals the
projection to $U$. The preimage of a point $p\,\in\,M$ under $\pi$ is called
the fiber of the bundle over $p$, it is a submanifold
$\mathscr{F}_{p}M\,:=\,\pi^{-1}(\,p\,)\,\subset\,\mathscr{F}M$ of the total
space $\mathscr{F}M$ diffeomorphic to the model fiber $\mathscr{F}$. Morphisms
between two fiber bundles are smooth maps
$\varphi:\,\mathscr{F}M\longrightarrow\hat{\mathscr{F}}M$ between the total
spaces which commute with the respective projections

(1)
$\mathscr{F}M$$\hat{\mathscr{F}}M$$M$$\scriptstyle\varphi$$\scriptstyle\pi$$\scriptstyle\hat{\pi}$

and thus map the fibers of $\mathscr{F}M$ to the fibers of
$\hat{\mathscr{F}}M$ over the same point.

Intuitively, a non–linear or Ehresmann connection on a fiber bundle
$\mathscr{F}M$ is a left hand side section $\nabla$ of the short exact
sequence of vector bundles

(2) $0\;\longrightarrow\;\hbox
to0.0pt{\begin{picture}(112.0,27.0)(0.0,0.0)\qbezier(89.0,11.0)(66.0,22.0)(45.0,11.0)\put(45.0,11.0){\vector(-2,-1){0.0}}
\put(60.0,20.0){$\nabla$}
\end{picture}\hss}\mathrm{Vert}\;\mathscr{F}M\;\stackrel{{\scriptstyle\subset}}{{\longrightarrow}}\;T\,\mathscr{F}M\;\stackrel{{\scriptstyle\pi_{*}}}{{\longrightarrow}}\;\pi^{*}TM\;\longrightarrow\;0$

over $\mathscr{F}M$, the corresponding right hand side section is
appropriately called the horizontal lift associated to $\nabla$. In order to
studying connections we prefer the following version of this intuitive notion
of a non–linear connection:

######  Definition 2.1 (Non–linear Connections on Fiber Bundles).

A general or a non–linear connection on a fiber bundle $\mathscr{F}M$ over a
manifold $M$ is a field $\mathbb{P}^{\nabla}$ $\in$
$\Gamma(\,\mathscr{F}M,\,\mathrm{End}\,T\mathscr{F}M\,)$ of projections
$(\mathbb{P}^{\nabla})^{2}\,=\,\mathbb{P}^{\nabla}$ on the tangent bundle
$T\mathscr{F}M$ such that its image distribution equals the vertical
foliation:

$\mathrm{im}\;\Big{(}\;\mathbb{P}^{\nabla}_{f}:\;\;T_{f}\mathscr{F}M\;\longrightarrow\;T_{f}\mathscr{F}M\;\Big{)}\;\;\stackrel{{\scriptstyle!}}{{=}}\;\;\mathrm{Vert}_{f}\mathscr{F}M\
.$

Every non–linear connection $\mathbb{P}^{\nabla}$ on a fiber bundle
$\mathscr{F}M$ allows us to define the directional derivative
$D^{\nabla}_{X}f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,\mathrm{Vert}\,\mathscr{F}M\,)$
of a given local section $f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,\mathscr{F}M\,)$
in the direction of a vector field $X$ on $M$ by:

$(\;D^{\nabla}_{X}f\;)_{p}\;\;:=\;\;\Big{(}\;T_{p}M\;\stackrel{{\scriptstyle
f_{*,\,p}}}{{\longrightarrow}}\;T_{f(p)}\mathscr{F}M\;\stackrel{{\scriptstyle\mathbb{P}^{\nabla}_{f(p)}}}{{\longrightarrow}}\;\mathrm{Vert}_{f(p)}\mathscr{F}M\;\Big{)}\;X_{p}\
.$

These directional derivatives assemble into a first order differential
operator

(3)
$D^{\nabla}:\;\;\Gamma(\,M,\,TM\,)\;\times\;\Gamma_{\mathrm{loc}}(\,M,\,\mathscr{F}M\,)\;\longrightarrow\;\Gamma_{\mathrm{loc}}(\,M,\,\mathrm{Vert}\,\mathscr{F}M\,)\
,$

which is the non–linear analogue of the classical definition of covariant
derivatives on vector bundles. Somewhat annoyingly this covariant derivative
$D^{\nabla}_{X}f$ contains the redundant information
$f\,=\,\pi_{\mathscr{F}M}\,\circ\,D^{\nabla}_{X}f$, where $\pi_{\mathscr{F}M}$
denotes the vertical tangent bundle projection
$\mathrm{Vert}\,\mathscr{F}M\longrightarrow\mathscr{F}M$. The simplicity of
linear and principal connections stems from the fact that we can get rid of
this redundancy altogether, the reduced covariant derivative $\nabla_{X}f$
captures only the partial derivatives of the section $f$.

The Nijenhuis or curvature tensor of a non–linear connection
$\mathbb{P}^{\nabla}$ on a fiber bundle $\mathscr{F}M$ over a manifold $M$ is
the horizontal $2$–form $R^{\nabla}$ on the total space $\mathscr{F}M$ of the
fiber bundle with values in the vertical tangent bundle defined for two
arbitrary vector fields $X,\,Y$ on $\mathscr{F}M$ by:

$\displaystyle R^{\nabla}(\;X,\,Y\;)$ $\displaystyle:=$
$\displaystyle-\;\mathbb{P}^{\nabla}\,[\;(\,\mathrm{id}-\mathbb{P}^{\nabla}\,)\,X,\;(\,\mathrm{id}-\mathbb{P}^{\nabla}\,)\,Y\;]\;-\;(\,\mathrm{id}-\mathbb{P}^{\nabla}\,)\,[\;\mathbb{P}^{\nabla}X,\;\mathbb{P}^{\nabla}Y\;]$
$\displaystyle=$
$\displaystyle-\,\mathbb{P}^{\nabla}\,[\,X,\,Y\,]\,+\,\mathbb{P}^{\nabla}\,[\,X,\,\mathbb{P}^{\nabla}Y\,]\,+\,\mathbb{P}^{\nabla}\,[\,\mathbb{P}^{\nabla}X,\,Y\,]\,-\,[\,\mathbb{P}^{\nabla}X,\mathbb{P}^{\nabla}Y\,]\
.$

The strange sign is necessary to make this definition agree with the classical
definition of the curvature of a linear connection, compare equation (8). By
construction
$R^{\nabla}\,\in\,\Omega^{2}_{\mathrm{hor}}(\,\mathscr{F}M,\,\mathrm{Vert}\,\mathscr{F}M\,)$
is $C^{\infty}(\,\mathscr{F}M\,)$–bilinear and alternating. The integrability
constraint for the vertical foliation $\mathrm{Vert}\,\mathscr{F}M$ tells us
$(\,\mathrm{id}\,-\,\mathbb{P}^{\nabla}\,)\,[\,\mathbb{P}^{\nabla}X,\,\mathbb{P}^{\nabla}Y\,]\,=\,0$,
and the simplified expression

(4)
$R^{\nabla}(\;X,\;Y\;)\;\;=\;\;-\;\mathbb{P}^{\nabla}\,[\;(\,\mathrm{id}\,-\,\mathbb{P}^{\nabla}\,)\,X,\;(\,\mathrm{id}\,-\,\mathbb{P}^{\nabla}\,)\,Y\;]\
,$

evidently results in a vertical vector field for all vector fields $X,\,Y$ on
$\mathscr{F}M$ and vanishes for a vertical vector field $X$ or $Y$ no matter
what is the other. In particular the curvature $R^{\nabla}$ measures exactly
the failure of the horizontal distribution
$\ker\,\mathbb{P}^{\nabla}\,\subseteq\,T\mathscr{F}M$ associated to
$\mathbb{P}^{\nabla}$ to be integrable.

An interpretation of the curvature tensor along classical lines as a
commutator of covariant derivatives requires a significant amount of extra
work and so we will only sketch the basic idea. The iterated tangent bundle of
a smooth manifold $\mathscr{F}$ is a smooth manifold $T(\,T\mathscr{F}\,)$
coming along with a canonical involutive diffeomorphism
$\Theta:\,T(\,T\mathscr{F}\,)\longrightarrow T(\,T\mathscr{F}\,)$
characterized by

(5)
$\Theta\;\left(\;\left.\frac{d}{dt}\right|_{0}\;\left.\frac{d}{d\varepsilon}\right|_{0}\gamma(\,t,\,\varepsilon\,)\;\right)\;\;=\;\;\left(\;\left.\frac{d}{dt}\right|_{0}\;\left.\frac{d}{d\varepsilon}\right|_{0}\gamma(\,\varepsilon,\,t\,)\;\right)$

for all $\gamma:\,\mathbb{R}^{2}\longrightarrow\mathscr{F}$. By construction,
$\Theta$ fits into the commutative diagram

(6) $T(\,T\mathscr{F}\,)$$T(\,T\mathscr{F}\,)$$T\mathscr{F}\oplus
T\mathscr{F}$$T\mathscr{F}\oplus
T\mathscr{F}$$\scriptstyle\Theta$$\scriptstyle\mathrm{swap}$$\scriptstyle\Pi$$\scriptstyle\Pi$

where $\mathrm{swap}$ interchanges the two summands and $\Pi$ is the double
projection:

$\Pi\left(\;\left.\frac{d}{dt}\right|_{0}\;\left.\frac{d}{d\varepsilon}\right|_{0}\gamma(\,t,\varepsilon\,)\;\right)\;\;:=\;\;\left.\frac{d}{d\varepsilon}\right|_{0}\gamma(\,0,\,\varepsilon\,)\;\oplus\;\left.\frac{d}{dt}\right|_{0}\gamma(\,t,\,0\,)\
.$

The involution $\Theta$ and double projection $\Pi$ generalize directly to
iterated vertical tangent bundles, because we may identify the fibers of
$\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M$ considered as a fiber bundle over
$M$ in every point $p\,\in\,M$ with the iterated tangent bundle
$T(\,T[\,\mathscr{F}_{p}M\,]\,)\,=\,[\,\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M\,]_{p}$.
The involutions and double projections thus defined on all the fibers of
$\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M$ assemble into an involutive fiber
bundle automorphism $\Theta$ and a double projection
$\Pi:\,\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M\longrightarrow\mathrm{Vert}\,\mathscr{F}M\,\oplus\,\mathrm{Vert}\,\mathscr{F}M$.
Making good use of the involution $\Theta$ we extend a non–linear connection
$\mathbb{P}^{\nabla}$ on $\mathscr{F}M$ to a unique non–linear connection
$\mathbb{P}^{\nabla^{\mathrm{Vert}}}$ on $\mathrm{Vert}\,\mathscr{F}M$ by
stipulating the identity

(7)
$D^{\nabla^{\mathrm{Vert}}}_{X}\Big{(}\;\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}\;\Big{)}\;\;=\;\;\Theta\,\Big{(}\;\left.\frac{d}{d\varepsilon}\right|_{0}D^{\nabla}_{X}f_{\varepsilon}\;\Big{)}$

for every vector field $X\,\in\,\Gamma(\,M,\,TM\,)$ and every smooth
one–parameter family $(\,f_{\varepsilon}\,)_{\varepsilon\,\in\,\mathbb{R}}$ of
local sections $f_{\varepsilon}$ of $\mathscr{F}M$ with infinitesimal
variation
$\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,\mathrm{Vert}\,\mathscr{F}M\,)$.
In this construction of the induced connection on
$\mathrm{Vert}\,\mathscr{F}M$, the involution $\Theta$ is proper remedy for
the nuisance

$\begin{array}[]{ccccccl}\Pi\,\Big{(}\\!\\!\\!&D^{\nabla^{\mathrm{Vert}}}_{X}(\,\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}\,)&\\!\\!\\!\Big{)}\;\;=\;\;\Big{(}\\!\\!\\!&\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}&\\!\\!\\!\Big{)}\,\oplus\,\Big{(}\\!\\!\\!&D^{\nabla}_{X}f_{0}&\\!\\!\\!\Big{)}\\\\[7.0pt]
\Pi\,\Big{(}\\!\\!\\!&\left.\frac{d}{d\varepsilon}\right|_{0}D^{\nabla}_{X}f_{\varepsilon}&\\!\\!\\!\Big{)}\;\;=\;\;\Big{(}\\!\\!\\!&D^{\nabla}_{X}f_{0}&\\!\\!\\!\Big{)}\,\oplus\,\Big{(}\\!\\!\\!&\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}&\\!\\!\\!\Big{)}\
,\end{array}$

compare the commutative diagram (6). For essentially the same reason the
involution $\Theta$ appears in the key identity linking the curvature
$R^{\nabla}$ of a non–linear connection $\mathbb{P}^{\nabla}$ on a fiber
bundle $\mathscr{F}M$ over $M$ to the commutator

(8)
$R^{\nabla}_{X,\,Y}f\;\;=\;\;\Big{(}\;D^{\nabla^{\mathrm{Vert}}}_{X}D^{\nabla^{\hphantom{V}}}_{Y}\\!\\!f\;-\;\Theta(\;D^{\nabla^{\mathrm{Vert}}}_{Y}D^{\nabla^{\hphantom{V}}}_{X}\\!\\!f\;)\;\Big{)}\;-\;D^{\nabla^{\hphantom{V}}}_{[\,X,\,Y\,]}f$

of covariant derivatives of a local section
$f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,\mathscr{F}M\,)$ in the direction of
vector fields $X,\,Y\,\in\,\Gamma(\,M,\,TM\,)$, in which $R^{\nabla}_{X,\,Y}f$
is a simplified notation for the local section of
$\mathrm{Vert}\,\mathscr{F}M$ over $M$ defined in $p\,\in\,M$ by:

$\Big{(}\;R^{\nabla}_{X,\,Y}f\;\Big{)}_{p}\;\;:=\;\;R^{\nabla}_{f(p)}(\;f_{*,\,p}X_{p},\;f_{*,\,p}Y_{p}\;)\;\;\in\;\;\mathrm{Vert}_{f(p)}\mathscr{F}M\
.$

The basic tenet of symbolic calculus on jet bundles provides the correct
interpretation for the differences in (8): The fibers of the double projection
$\Pi:\,\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M\longrightarrow\mathrm{Vert}\,\mathscr{F}M\oplus\mathrm{Vert}\,\mathscr{F}M$
are affine spaces modelled on the fibers of $\mathrm{Vert}\,\mathscr{F}M$ and
thus come along with a difference map

$-\;:\;\;\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M\;\times_{\mathrm{Vert}\,\mathscr{F}M\oplus\mathrm{Vert}\,\mathscr{F}M}\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M\;\longrightarrow\;\mathrm{Vert}\,\mathscr{F}M\
,$

which determines the inner difference in the curvature identity (8), the outer
difference is simply the difference in $\mathrm{Vert}\,\mathscr{F}M$
considered as a vector bundle over $\mathscr{F}M$. Taking care with this
delicate interpretation of differences we will prove the curvature identity
(8) in local coordinates in Appendix A.

######  Definition 2.2 (Parallel morphisms between Fiber Bundles).

A parallel morphism between fiber bundles $\mathscr{F}M$ and
$\hat{\mathscr{F}}M$ over the same manifold $M$ endowed with connections
$\mathbb{P}^{\nabla}$ and $\mathbb{P}^{\hat{\nabla}}$ respectively is a
morphism $\varphi:\,\mathscr{F}M\longrightarrow\hat{\mathscr{F}}M$ of fiber
bundles such that the following diagram commutes:

The constraint $\hat{\pi}\,\circ\,\varphi\,=\,\pi$ characterizing morphisms of
fiber bundles readily implies

$\varphi_{*}(\,\mathrm{Vert}\,\mathscr{F}M\,)\,\subset\,\mathrm{Vert}\,\hat{\mathscr{F}}M,$

hence a morphism $\varphi$ of fiber bundles is parallel, if and only if
$\varphi_{*}$ maps the horizontal distribution of $\mathscr{F}M$ to the
horizontal distribution of $\hat{\mathscr{F}}M$:

$\varphi\;\textrm{\
parallel}\qquad\Longleftrightarrow\qquad\varphi_{*}(\;\ker\;\mathbb{P}^{\nabla}\;)\;\;\subset\;\;\ker\;\mathbb{P}^{\hat{\nabla}}\
.$

##  3\. Principal Connections

Having discussed general non–linear connections on fiber bundles in some
detail we now want to specialize to principal connections on principal bundles
in this section. Recall first of all that a principal bundle modelled on a Lie
group $G$ is a smooth fiber bundle $GM$ with model fiber $G$ endowed with a
smooth right, fiber preserving action of $G$ on its total space $GM$

$\rho\,:\;\;GM\;\times\;G\;\longrightarrow\;GM,\qquad(\,g\,,\,\gamma\,)\;\longmapsto\;g\,\gamma\
,$

which is simply transitive on all fibers of $GM$ over $M$. Every local section
${\scriptstyle\Gamma}:\,U\longrightarrow GM$ of the projection
$\pi:\,GM\longrightarrow M$ defined over an open subset $U$ extends to a
$G$–equivariant local trivialization of $GM$ over $U$

(9) $\Phi:\;\;U\times
G\;\stackrel{{\scriptstyle{\scriptscriptstyle\Gamma}\times\mathrm{id}}}{{\longrightarrow}}\;\pi^{-1}(\,U\,)\times
G\;\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}\;\pi^{-1}(\,U\,)\ ,$

satisfying
$\Phi(u,g\gamma)\,=\,{\scriptstyle\Gamma}(u)g\gamma\,=\,\Phi(u,g)\,\gamma$ by
construction. The existence of $G$–equivariant trivializations ensures that
the unique set theoretic map

(10)
$\backslash\;:\;\;GM\;\times_{M}\,GM\;\longrightarrow\;G,\qquad(\,g,\,\hat{g}\,)\;\longmapsto\;g^{-1}\,\hat{g}\
,$

satisfying $g\,(g^{-1}\hat{g})\,=\,\hat{g}$ for all $g,\,\hat{g}\,\in\,GM$ in
the same fiber is actually a smooth map, because it reads

$\Phi(u,g)^{-1}\Phi(u,\hat{g})\,=\,g^{-1}\hat{g}$

in every such trivialization $\Phi$.

The automorphism group bundle of a principal bundle $GM$ over a manifold $M$
is the Lie group bundle $\mathrm{Aut}\;GM$ over $M$ defined by

(11)
$\mathrm{Aut}\,GM\;\;:=\;\;\\{\;\;(p,\psi)\;\;|\;\;\psi:\,G_{p}M\longrightarrow
G_{p}M\textrm{\ is $G$--equivariant}\;\;\\}$

with the bundle projection
$\pi_{\mathrm{Aut}\,GM}\,:\,\mathrm{Aut}\,GM\longrightarrow
M,\,(\,p,\,\psi\,)\longmapsto p$. In mathematical physics, the Fréchet–Lie
group $\Gamma(\,M,\,\mathrm{Aut}\,GM\,)$ of all global sections of the
automorphism bundle is called the gauge group of $GM$.

The fiber of the Lie group bundle $\mathrm{Aut}\;GM$ over a point $p\,\in\,M$
is a Lie group $\mathrm{Aut}_{p}GM$ isomorphic, although not canonically so,
to the original group $G$, in particular its Lie algebra
$\mathfrak{aut}_{p}GM\,\cong\,\mathfrak{g}$ is isomorphic to the Lie algebra
of $G$. All these Lie algebras assemble into a smooth Lie algebra bundle
$\mathfrak{aut}\,GM$, whose global sections
$\Gamma(\,M,\,\mathfrak{aut}\,GM\,)$ form the Fréchet–Lie algebra of the gauge
group $\Gamma(\,M,\,\mathrm{Aut}\,GM\,)$ of the principal bundle $GM$.

######  Definition 3.1 (Principal Connections).

A principal connection on a principal $G$–bundle $GM$ over a manifold $M$ is a
non–linear connection $\mathbb{P}^{\nabla}$ on the fiber bundle $GM$, which is
invariant under the right action of $G$ on $GM$ in the sense that the right
translations $R_{\gamma}:\,GM\longrightarrow GM,\,g\longmapsto g\gamma,$ are
parallel automorphisms for all $\gamma\,\in\,G$.

In difference to general fiber bundles, the vertical tangent bundle of a
principal bundle $GM$ is trivializable: The construction of the Maurer–Cartan
form on a Lie group $G$ applies verbatim and results in the vertical
trivialization isomorphism

$\mathrm{Vert}\,GM\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}GM\times\mathfrak{g},\;\left.\frac{d}{dt}\right|_{0}g_{t}\longmapsto(g_{0},\left.\frac{d}{dt}\right|_{0}g^{-1}_{0}g_{t})$

under the proviso that the curve $t\longmapsto g_{t}$ chosen to represent the
vertical tangent vector stays in the fiber so that the curve $t\longmapsto
g_{0}^{-1}g_{t}$ is defined by application (10). The composition of this
vertical trivialization with the projection to $\mathfrak{g}$

(12)
$\mathrm{vtriv}\,:$$\mathrm{Vert}\,GM$$GM\times\mathfrak{g}$$GM$$\mathfrak{g}$$\scriptstyle\cong$$\scriptstyle\pi_{GM}$$\scriptstyle\mathrm{pr}_{L}$$\scriptstyle\mathrm{pr}_{R}$

induces isomorphisms
$\mathrm{vtriv}_{g}:\,\mathrm{Vert}_{g}GM\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\mathfrak{g}$
at every $g\,\in\,GM$ and thus a bijection
$\mathbb{P}^{\nabla}\,\longleftrightarrow\,\omega$ between non–linear
connections and $\mathfrak{g}$–valued $1$–forms
$\omega\,\in\,\Omega^{1}(\,GM,\,\mathfrak{g}\,)$, which agree with
$\mathrm{vtriv}$ on vertical tangent vectors:

(13)
$\omega\;\;:=\;\;\mathrm{vtriv}\,\circ\,\mathbb{P}^{\nabla}\qquad\Longleftrightarrow\qquad\mathbb{P}^{\nabla}\;\;:=\;\;\mathrm{vtriv}^{-1}\,\circ\,\omega\
.$

This description of non–linear connections on $GM$ by $\mathfrak{g}$–valued
$1$–forms $\omega$ with
$\left.\omega\right|_{\mathrm{Vert}\,GM}\,=\,\mathrm{vtriv}$ is much more
convenient and will be used for the rest of this article.

To characterize the principal connections specified in Definition 3.1 by their
connection forms $\omega$ we recall that the right translation by
$\gamma\,\in\,G$ is a parallel morphism

$R_{\gamma}:\,GM\longrightarrow GM,\;g\longmapsto g\gamma,$

if and only if its differential $R_{\gamma*}$ maps the horizontal vectors
$\ker\,\omega_{g}\,\subset\,T_{g}GM$ at a point $g\,\in\,GM$ to horizontal
vectors $\ker\omega_{g\gamma}\,\subset\,T_{g\gamma}GM$. On the complementary
vertical vectors the connection form behaves like
$\mathrm{vtriv}\,=\,\left.\omega\right|_{\mathrm{Vert}\,GM}$:

$(\mathrm{vtriv}\,\circ\,R_{\gamma*})\Big{(}\\!\left.\frac{d}{dt}\right|_{0}g_{t}\Big{)}\;\;=\;\;\left.\frac{d}{dt}\right|_{0}\\!(\gamma
g_{0})^{-1}(g_{t}\gamma)\;\;=\;\;\mathrm{Ad}_{\gamma^{-1}}\mathrm{vtriv}\Big{(}\\!\left.\frac{d}{dt}\right|_{0}g_{t}\Big{)}\
.$

In consequence the condition
$\omega\,\circ\,R_{\gamma*}\,=\,\mathrm{Ad}_{\gamma^{-1}}\,\circ\,\omega$ for
all $\gamma\,\in\,G$ is necessary and sufficient for a $\mathfrak{g}$–valued
$1$–form $\omega$ agreeing on vertical vectors with
$\mathrm{vtriv}\,=\,\left.\omega\right|_{\mathrm{Vert}\,GM}$ to be the
connection form of a principal connection $\mathbb{P}^{\nabla}$. The chain
rule in the guise of a Leibniz rule tells us for such a form

$\displaystyle\omega_{g_{0}\gamma_{0}}\Big{(}\;\left.\frac{d}{dt}\right|_{0}g_{t}\,\gamma_{t}\;\Big{)}$
$\displaystyle=$
$\displaystyle\omega_{g_{0}\gamma_{0}}\Big{(}\;\left.\frac{d}{dt}\right|_{0}g_{t}\,\gamma_{0}\;+\;\left.\frac{d}{dt}\right|_{0}g_{0}\,\gamma_{t}\;\Big{)}$
$\displaystyle=$
$\displaystyle\omega_{g_{0}\gamma_{0}}\Big{(}\;(\,R_{\gamma_{0}}\,)_{*}\left.\frac{d}{dt}\right|_{0}g_{t}\;\Big{)}\;+\;\left.\frac{d}{dt}\right|_{0}(g_{0}\gamma_{0})^{-1}(g_{0}\gamma_{t})$
$\displaystyle=$
$\displaystyle\mathrm{Ad}_{\gamma^{-1}_{0}}\;\omega_{g_{0}}\Big{(}\;\left.\frac{d}{dt}\right|_{0}g_{t}\;\Big{)}\;+\;\left.\frac{d}{dt}\right|_{0}\gamma_{0}^{-1}\gamma_{t}$

for every curve $t\longmapsto g_{t}$ in $GM$ and $t\longmapsto\gamma_{t}$ in
$G$. Put differently:

######  Lemma 3.2 (Principal Connection Axiom).

On every principal $G$–bundle $GM$ the association
$\mathbb{P}^{\nabla}\,\longleftrightarrow\,\omega$ characterized by
$\omega\,:=\,\mathrm{vtriv}\circ\mathbb{P}^{\nabla}$ induces a bijection
between principal connections in the sense of Definition 3.1 and
$\mathfrak{g}$–valued $1$–forms $\omega$ on $GM$ satisfying the axiom

$\omega_{g_{0}\gamma_{0}}\Big{(}\;\left.\frac{d}{dt}\right|_{0}g_{t}\,\gamma_{t}\;\Big{)}\;\;=\;\;\mathrm{Ad}_{\gamma^{-1}_{0}}\;\omega_{g_{0}}\Big{(}\;\left.\frac{d}{dt}\right|_{0}g_{t}\;\Big{)}\;+\;\left.\frac{d}{dt}\right|_{0}\gamma_{0}^{-1}\gamma_{t}$

for all choices of smooth curves $t\longmapsto g_{t}$ in $GM$ and curves
$t\longmapsto\gamma_{t}$ in $G$.

Cartan’s Second Structure Equation [1] is a convenient description of the
image of the composition of the curvature tensor $R^{\nabla}$ with the
vertical trivialization $\mathrm{vtriv}$ in terms of the exterior derivative
of the connection form

(14)
$\Omega\;\;:=\;\;\mathrm{vtriv}\,\circ\,R^{\nabla}\;\;\stackrel{{\scriptstyle!}}{{=}}\;\;d\omega\;+\;\frac{1}{2}\,[\,\omega\,\wedge\,\omega\,]\
,$

where
$\frac{1}{2}[\,\omega\wedge\omega\,](X,Y)\,:=\,[\,\omega(X),\,\omega(Y)\,]$;
the usefulness of the factor $\frac{1}{2}$ in this definition is beyond
question. Cartan’s Second Structure Equation can be easily verified directly.
On the other hand it can be derived without too much additional effort from
the curvature identity (8) by considering the reduced covariant derivative
associated to a principal connection $\omega$

$\nabla:\;\;\Gamma(\,M,\,TM\,)\,\times\,\Gamma_{\mathrm{loc}}(\,M,\,GM\,)\;\longrightarrow\;C^{\infty}_{\mathrm{loc}}(\,M,\,\mathfrak{g}\,)$

defined by $\nabla_{X}f\,:=\,\mathrm{vtriv}(\,D^{\nabla}_{X}f\,)$; in light of
equation (13) this becomes

$(\;\nabla_{X}f\;)_{p}\;\;:=\;\;\mathrm{vtriv}(\;\mathbb{P}^{\nabla}_{f(p)}(\,f_{*,\,p}X_{p}\,)\;)\;\;\stackrel{{\scriptstyle!}}{{=}}\;\;(\,f^{*}\omega\,)_{p}(\,X\,)$

for all $X\,\in\,\Gamma(M,TM)$. Iterating the vertical trivialization
isomorphism $\mathrm{Vert}\,GM\,\longrightarrow GM\times\mathfrak{g}$ used to
define $\mathrm{vtriv}$ we obtain the isomorphism:

(15)
$\Psi:\;\mathrm{Vert}\,\mathrm{Vert}\,GM\;\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\;\mathrm{Vert}(\,GM\times\mathfrak{g}\,)\;\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\;(GM\times\mathfrak{g})\,\times\,(\mathfrak{g}\times\mathfrak{g})\
.$

Its inverse can be written down explicitly by unravelling the definitions

$\displaystyle\Psi^{-1}(\,g,\,X,\,Y,\,Z\,)$ $\displaystyle=$
$\displaystyle\left.\frac{d}{dt}\right|_{0}\,\left.\frac{d}{d\varepsilon}\right|_{0}g\,e^{tX}\,e^{\varepsilon(\,Y\,+\,tZ\,)}$
$\displaystyle=$
$\displaystyle\left.\frac{d}{dt}\right|_{0}\,\left.\frac{d}{d\varepsilon}\right|_{0}g\,e^{tX\,+\,\varepsilon
Y\,+\,t\varepsilon(\,Z\,+\,\frac{1}{2}\,[X,Y]\,)}\ ,$

the second equality is the Baker–Campbell–Hausdorff Formula [2] in the Lie
group $G$ ignoring as usual terms of order $O(\,t^{2},\,\varepsilon^{2}\,)$.
In particular the involution $\Theta$ defined in equation (5) simply by the
interchange $t\,\leftrightarrow\,\varepsilon$ is not as innocent as it may
appear, under the isomorphism $\Psi$ it becomes:

(16)
$\Theta(\;g,\;X;\;Y,\;Z\;)\;\;\stackrel{{\scriptstyle!}}{{=}}\;\;(\;g,\;Y;\;X,\;Z\,+\,[\,X,\,Y\,]\;)\
.$

In accordance with [3] the vertical trivialization isomorphism
$\mathrm{Vert}\,GM\longrightarrow GM\times\mathfrak{g}$ is parallel with
respect to the connections $\mathbb{P}^{\nabla^{\mathrm{Vert}}}$ and
$\mathbb{P}^{\nabla}\oplus\mathbb{P}^{\mathrm{triv}}$ on $\mathrm{Vert}\,GM$
and $GM\times\mathfrak{g}$ respectively, using the fiber bundle isomorphism
$\Psi$ we thus obtain

$\displaystyle\Psi(\;D^{\nabla^{\mathrm{Vert}}}_{X}\,D^{\nabla^{\hphantom{V}}}_{Y}\\!\\!f\;)$
$\displaystyle=$
$\displaystyle(\;f,\;(f^{*}\omega)(X);\;(f^{*}\omega)(Y),\;X\,(f^{*}\omega)(Y)\;)$
$\displaystyle\Psi(\;D^{\nabla^{\mathrm{Vert}}}_{Y}\,D^{\nabla^{\hphantom{V}}}_{X}\\!\\!f\;)$
$\displaystyle=$
$\displaystyle(\;f,\;(f^{*}\omega)(Y);\;(f^{*}\omega)(X),\;Y\,(f^{*}\omega)(X)\;)$

for all local sections $f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,GM\,)$ and all
$X,\,Y\,\in\,\Gamma(\,M,\,TM\,)$. Evidently the double projections under $\Pi$
of these two local sections of $\mathrm{Vert}\,\mathrm{Vert}\,GM$ do not agree
unless we apply the involution $\Theta$ first picking up the rather unexpected
additional term of equation (16) along the way:

$\displaystyle
D^{\nabla^{\mathrm{Vert}}}_{X}\,D^{\nabla^{\hphantom{V}}}_{Y}\\!\\!f\;-\;\Theta(\;D^{\nabla^{\mathrm{Vert}}}_{Y}\,D^{\nabla^{\hphantom{V}}}_{X}\\!\\!f\;)$
$\displaystyle=$
$\displaystyle\Big{(}\;f,\;X(f^{*}\omega)(Y)\;-\;Y(f^{*}\omega)(X)\;-\;[\,(f^{*}\omega)(Y),\,(f^{*}\omega)(X)\,]\;\Big{)}$

Subtracting $D^{\nabla}_{[X,Y]}f\,=\,(\,f,\,(f^{*}\omega)([X,Y])\,)$ we obtain
eventually

$\displaystyle R^{\nabla}_{X,\,Y}f$ $\displaystyle=$
$\displaystyle(\;f,\;d(f^{*}\omega)(X,Y)\;+\;\frac{1}{2}[\,(f^{*}\omega)\,\wedge\,(f^{*}\omega)\,](\,X,\,Y\,)\;)$
$\displaystyle=$ $\displaystyle(\;f,\;(\,f^{*}\Omega\,)(\,X,\,Y\,)\;)$

for all local sections $f\,\in\,\Gamma_{\mathrm{loc}}(\,M,GM\,)$ and all
$X,\,Y\,\in\,\Gamma(\,M,TM\,)$ by using the naturality
$d(f^{*}\omega)\,=\,f^{*}(d\omega)$ of the exterior derivative $d$ and the
definition of the $2$–form $\Omega$ in Cartan’s Second Structure Equation
(14).

##  4\. Linear Connections

The strategy persued for linear connections on vector bundles $VM$ follows the
model of principal connections closely. The tangent bundle of a vector space
is canonically trivializable $TV\,\cong\,V\times V$ by taking actual
derivatives and this becomes via
$[\,\mathrm{Vert}\,VM\,]_{p}\,=\,T(\,V_{p}M\,)$ the vertical trivialization

$\mathrm{Vert}\,VM\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}VM\oplus
VM,\;\left.\frac{d}{dt}\right|_{0}v_{t}\longmapsto v_{0}\oplus\lim_{t\to
0}\frac{1}{t}(v_{t}-v_{0}),$

again under the proviso that the representative curve $t\longmapsto v_{t}$ is
chosen to stay in the same vector space so that the quotient
$\frac{1}{t}(v_{t}-v_{0})\,\in\,V_{\pi(v_{0})}M$ is well–defined. Composing
with the projection to the right factor we obtain

(17) $\mathrm{vtriv}:$$\mathrm{Vert}\,VM$$VM\oplus
VM$$VM$$VM$,$\scriptstyle\cong$$\scriptstyle\pi_{VM}$$\scriptstyle\mathrm{pr}_{L}$$\scriptstyle\mathrm{pr}_{R}$

which can be used to project out
$\nabla_{X}v\,:=\,\mathrm{vtriv}(D^{\nabla}_{X}v)$ the redundant information
from the covariant derivative $D^{\nabla}_{X}v$ of a section
$v\,\in\,\Gamma(\,M,\,VM\,)$:

######  Definition 4.1 (Linear Connections on Vector Bundles).

A linear connection on a vector bundle $VM$ on $M$ is a non–linear connection
$\mathbb{P}^{\nabla}$ on $VM$ such that the reduced covariant derivative is
$\mathbb{R}$–bilinear:

$\nabla:\;\;\Gamma(\,M,\,TM\,)\;\times\;\Gamma(\,M,\,VM\,)\;\longrightarrow\;\Gamma(\,M,\,VM\,)\
.$

In order to characterize linear connections by a condition easier to verify we
consider a vector bundle $VM$ with projection $\pi:\,VM\longrightarrow M$ and
a local basis of sections
$\mathbf{v}_{1},\,\ldots,\,\mathbf{v}_{n}\,\in\,\Gamma_{\mathrm{loc}}(\,M,\,VM\,)$
defined over the open domain $U\,\subseteq\,M$ of a local coordinate chart
$x:\,U\longrightarrow x(U)\,\subseteq\,\mathbb{R}^{m}$. In this situation we
can construct linear local coordinates for the total space $VM$ of the vector
bundle $(x,v):\,\pi^{-1}(U)\longrightarrow x(U)\times\mathbb{R}^{n}$
characterized by:

$(\,x^{1},\,\ldots\,x^{m};\,v^{1},\,\ldots,\,v^{n}\,)\;\;\longleftrightarrow\;\;v^{1}\,\mathbf{v}_{1}\;+\;\ldots\;+\;v^{n}\,\mathbf{v}_{n}\;\;\in\;\;V_{(x^{1},\ldots,x^{m})}M\
.$

In these linear local coordinates a general non–linear connection
$\mathbb{P}^{\nabla}$ reads

(18) $\mathbb{P}^{\nabla}_{(x,v)}\Big{(}\frac{\partial}{\partial
v^{\alpha}}\Big{)}\;\;=\;\;\frac{\partial}{\partial
v^{\alpha}}\qquad\mathbb{P}^{\nabla}_{(x,v)}\Big{(}\frac{\partial}{\partial
x^{\mu}}\Big{)}\;\;=\;\;\sum_{\alpha\,=\,1}^{n}\Gamma^{\alpha}_{\mu}(\,x,\,v\,)\,\frac{\partial}{\partial
v^{\alpha}}$

with local coefficient functions $\Gamma^{\alpha}_{\mu}(\,x,\,v\,)$
generalizing the classical Christoffel symbols, because the coordinate vector
fields $\frac{\partial}{\partial v^{1}},\,\ldots,\,\frac{\partial}{\partial
v^{n}}$ span the vertical tangent bundle in every point of
$\pi^{-1}(\,U\,)\,\subseteq\,VM$.

With respect to the local trivialization of the vector bundle $VM$ provided by
the local basis $\mathbf{v}_{1},\,\ldots,\,\mathbf{v}_{n}$ every section
$v\,\in\,\Gamma(\,M,\,VM\,)$ can be expanded as a sum
$v\,=\,v^{1}\mathbf{v}_{1}\,+\,\ldots\,+\,v^{n}\mathbf{v}_{n}$ with local
coefficient functions $v^{1},\,\ldots,\,v^{n}$ in $C^{\infty}(U)$, which of
course reappear in the local coordinate description of the section
$v\,\in\,\Gamma(\,M,\,VM\,)$ considered as a smooth map:

$(\,x^{1},\,\ldots,\,x^{m}\,)\;\longmapsto\;(\,x^{1},\,\ldots,\,x^{m};\,v^{1}(x^{1},\ldots,x^{m}),\,\ldots,\,v^{n}(x^{1},\ldots,x^{m})\,)\
.$

Calculating the differential of $v$ in these coordinates is straightforward:

$\displaystyle\mathbb{P}^{\nabla}\left(v_{*}\,\Big{(}\,\frac{\partial}{\partial
x^{\mu}}\,\Big{)}\right)$ $\displaystyle=$
$\displaystyle\mathbb{P}^{\nabla}\Big{(}\;\frac{\partial}{\partial
x^{\mu}}\;+\;\sum_{\alpha\,=\,1}^{n}\frac{\partial v^{\alpha}}{\partial
x^{\mu}}(\,x\,)\,\frac{\partial}{\partial v^{\alpha}}\;\Big{)}$
$\displaystyle=$ $\displaystyle\sum_{\alpha\,=\,1}^{n}\Big{(}\;\frac{\partial
v^{\alpha}}{\partial
x^{\mu}}(\,x\,)\;+\;\Gamma_{\mu}^{\alpha}(\,x,\,v(x)\,)\;\Big{)}\;\frac{\partial}{\partial
v^{\alpha}}\ .$

Pushing this expression to a section of $VM$ via the vertical trivialization
$\mathrm{vtriv}$ in diagram (17) converts the coordinate vector field
$\frac{\partial}{\partial v^{\alpha}}$ back to the corresponding local section
$\mathbf{v}^{\alpha}$, hence it reduces the covariant derivative to:

(19) $\nabla_{\frac{\partial}{\partial
x^{\mu}}}v\;\;:=\;\;\sum_{\alpha\,=\,1}^{n}\Big{(}\;\frac{\partial
v^{\alpha}}{\partial
x^{\mu}}(\,x\,)\;+\;\Gamma_{\mu}^{\alpha}(\,x,\,v(x)\,)\;\Big{)}\;\mathbf{v}_{\alpha}\
.$

Directly from this expansion we conclude that a connection
$\mathbb{P}^{\nabla}$ on $VM$ will not be a linear connection in the sense of
Definition 4.1 unless its Christoffel symbols
$\Gamma^{\beta}_{\mu}(\,x,\,v\,)$ are linear in the coefficient coordinates
with an expansion

(20)
$\Gamma^{\alpha}_{\mu}(\,x^{1},\,\ldots,\,x^{m};\,v^{1},\,\ldots,\,v^{n}\,)\;\;=\;\;\sum_{\omega\,=\,1}^{n}\Gamma^{\alpha}_{\mu\omega}(\,x^{1},\,\ldots,\,x^{m}\,)\,v^{\omega}$

and suitable local coefficient functions $\Gamma^{\alpha}_{\mu\omega}(\,x\,)$.
In the chosen linear coordinates the scalar multiplication
$\Lambda_{\lambda}:\,VM\longrightarrow VM,\,v\longmapsto\lambda v,$ with
$\lambda\,\in\,\mathbb{R}$ reads
$(\,x^{1},\ldots,x^{m};\,v^{1},\ldots,v^{n}\,)\longmapsto(\,x^{1},\ldots,x^{m};\,\lambda
v^{1},\ldots,\lambda v^{n}\,)$. It is thus a parallel endomorphism of $VM$
with respect to the connection $\mathbb{P}^{\nabla}$, if and only if we have
for all $\mu$ equality between the two tangent vector expressions

$\displaystyle(\,\Lambda_{\lambda}\,)_{*,\,(x,v)}\Big{(}\,\frac{\partial}{\partial
x^{\mu}}-\sum_{\alpha\,=\,1}^{n}\Gamma^{\alpha}_{\mu}(\,x,\,v\,)\frac{\partial}{\partial
v^{\alpha}}\,\Big{)}$ $\displaystyle=$ $\displaystyle\frac{\partial}{\partial
x^{\mu}}-\sum_{\alpha\,=\,1}^{n}\lambda\;\Gamma^{\alpha}_{\mu}(\,x,\,v\,)\frac{\partial}{\partial
v^{\alpha}}\;\;\stackrel{{\scriptstyle?}}{{=}}\;\;\frac{\partial}{\partial
x^{\mu}}-\sum_{\alpha\,=\,1}^{n}\Gamma^{\alpha}_{\mu}(\,x,\,\lambda
v\,)\frac{\partial}{\partial v^{\alpha}}$

due to $(\,\Lambda_{\lambda}\,)_{*}\frac{\partial}{\partial
x^{\mu}}\,=\,\frac{\partial}{\partial x^{\mu}}$ and
$(\,\Lambda_{\lambda}\,)_{*}\frac{\partial}{\partial
v^{\alpha}}\,=\,\lambda\,\frac{\partial}{\partial v^{\alpha}}$. Note that the
tangent vectors on the left and right hand side of this equation are
horizontal in the points $(\,x,\,v\,)$ and $(\,x,\,\lambda v\,)$ of $VM$
respectively.

In consequence the scalar multiplications $\Lambda_{\lambda}$ are parallel
endomorphisms of $VM$ for all scalars $\lambda\,\in\,\mathbb{R}$, if and only
if all the generalized Christoffel symbols $\Gamma^{\alpha}_{\mu}(x,v)$ are
homogeneous functions of degree one in the coefficient coordinates
$v^{1},\,\ldots,\,v^{n}$. On the other hand every smooth homogeneous function
of degree one is actually a linear function, hence we have proved:

######  Lemma 4.2 (Characterization of Linear Connections).

A non–linear connection $\mathbb{P}^{\nabla}$ on a vector bundle gives rise to
an $\mathbb{R}$–bilinear covariant derivative
$\nabla:\,\Gamma(\,M,\,TM\,)\times\Gamma(\,M,\,VM\,)\longrightarrow\Gamma(\,M,\,VM\,)$,
if and only if the multiplication by every $\lambda\,\in\,\mathbb{R}$ is a
parallel endomorphism:

$\Lambda_{\lambda}:\;\;VM\;\longrightarrow\;VM,\qquad
v\;\longmapsto\;\lambda\,v\ .$

##  Appendix A Explicit Calculations in Local Coordinates

In order to illustrate the rather abstract and functorial discussion of
non–linear connections and their curvature in this article we will repeat
parts of the arguments of Section $2$ in this appendix by explicit
calculations in local coordinates. In particular we will use these
calculations in local coordinates to prove the general curvature identity (8)
relating the Nijenhuis tensor definition (4) of the curvature $R^{\nabla}$ of
a non–linear connection on a fiber bundle $\mathscr{F}M$ to the commutator of
iterated covariant derivatives of its sections.

For actual calculations the most convenient definition of the tangent bundle
of a smooth manifold $\mathscr{F}$ is as a set
$T\mathscr{F}\,:=\,\mathrm{Jet}^{1}_{0}(\,\mathbb{R},\,\mathscr{F}\,)$ of
equivalence classes of curves $\gamma:\,\mathbb{R}\longrightarrow\mathscr{F}$
under first order contact in $0\,\in\,\mathbb{R}$. The second iterated tangent
bundle can be defined analogously as the set $T(\,T\mathscr{F}\,)$ of
equivalence classes of maps
$\gamma:\,\mathbb{R}^{2}\longrightarrow\mathscr{F},\,(t,\varepsilon)\longmapsto\gamma(t,\varepsilon),$
under a modified equivalence relation of second order contact in $(0,0)$: Two
maps $\gamma:\,\mathbb{R}^{2}\longrightarrow\mathscr{F}$ and
$\hat{\gamma}:\,\mathbb{R}^{2}\longrightarrow\mathscr{F}$ determine the same
equivalence class

(21)
$\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}\gamma(\,t,\,\varepsilon\,)\;\;=\;\;\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}\hat{\gamma}(\,t,\,\varepsilon\,)\;\;\in\;\;T(\,T\mathscr{F}\,)\
,$

if and only if for some and hence every local coordinate chart
$(\,f^{1},\,\ldots,\,f^{n}\,)$ of $\mathscr{F}$ defined in a neighborhood of
$\gamma(\,0,0\,)\,=\,\hat{\gamma}(\,0,0\,)$ the $4n$ real numbers

(22)
$\begin{array}[]{rcl}f^{\alpha}\Big{(}\;{\displaystyle\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}}\gamma(\,t,\,\varepsilon\,)\;\Big{)}&=&\quad(\,f^{\alpha}\circ\gamma\,)(0,0)\\\\[10.0pt]
\dot{f}^{\alpha}\Big{(}\;{\displaystyle\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}}\gamma(\,t,\,\varepsilon\,)\;\Big{)}&=&\;\;{\displaystyle\left.\frac{\partial}{\partial
t}\;\;\right|_{(0,0)}}(\,f^{\alpha}\circ\gamma\,)\\\\[10.0pt]
\delta\\!f^{\alpha}\Big{(}\;{\displaystyle\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}}\gamma(\,t,\,\varepsilon\,)\;\Big{)}&=&\;\;{\displaystyle\left.\frac{\partial}{\partial\varepsilon}\;\;\right|_{(0,0)}}(\,f^{\alpha}\circ\gamma\,)\\\\[10.0pt]
\delta\\!\dot{f}^{\alpha}\Big{(}\;{\displaystyle\left.\frac{d}{dt}\right|_{0}\left.\frac{d}{d\varepsilon}\right|_{0}}\gamma(\,t,\,\varepsilon\,)\;\Big{)}&=&{\displaystyle\left.\frac{\partial^{2}}{\partial
t\,\partial\varepsilon}\right|_{(0,0)}}(\,f^{\alpha}\circ\gamma\,)\end{array}$

for $\gamma$ agree with the corresponding numbers for $\hat{\gamma}$. In this
case these $4n$ numbers are of course the coordinates of the common
equivalence class (21) in the induced coordinate chart
$(\,f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\,\delta\\!\dot{f}^{\alpha}\,)$
of $T(\,T\mathscr{F}\,)$. In contrast to $T\mathscr{F}$ the manifold
$T(\,T\mathscr{F}\,)$ is not a true jet space, the modification (22) of the
second order contact relation depends on the existence of tautological
coordinates $(\,t,\varepsilon\,)$ on $\mathbb{R}^{2}$. From the definition
(22) of the equivalence relation we see directly that the involution
$\Theta:\,T(T\mathscr{F})\longrightarrow T(T\mathscr{F})$ defined in equation
(5) by precomposing $\gamma:\,\mathbb{R}^{2}\longrightarrow\mathscr{F}$ with
the swap $(t,\,\varepsilon)\longmapsto(\varepsilon,\,t)$ is well–defined on
equivalence classes and takes the simple form

(23)
$\Theta\,(\;f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\,\delta\\!\dot{f}^{\alpha}\;)\;\;=\;\;(\;f^{\alpha};\,\delta\\!f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!\dot{f}^{\alpha}\;)$

in local coordinates, compare equation (16). Similarly the double projection
$\Pi:\,T(\,T\mathscr{F}\,)\longrightarrow T\mathscr{F}\oplus T\mathscr{F}$
becomes in the chosen local coordinates

$\Pi(\;f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\,\delta\\!\dot{f}^{\alpha}\;)\;\;=\;\;(\;f^{\alpha};\,\delta\\!f^{\alpha}\;)\oplus(\;f^{\alpha};\,\dot{f}^{\alpha}\;)\;\;=\;\;(\;f^{\alpha};\,\delta\\!f^{\alpha};\,\dot{f}^{\alpha}\;)\
,$

because $\oplus$ for vector bundles over $\mathscr{F}$ is essentially the same
as $\times_{\mathscr{F}}$ and effectively removes the redundant copy of the
coordinates $f^{\alpha}$. For an alternative coordinate chart
$(\,h^{1},\,\ldots,\,h^{n}\,)$ of the manifold $\mathscr{F}$ the chain rule
applied to the composition $h^{\omega}\circ\gamma\,=\,(h^{\omega}\circ
f^{-1})\circ(f\circ\gamma)$ tells us

$\delta\\!\dot{h}^{\omega}(\;f;\,\dot{f};\,\delta\\!f;\,\delta\\!\dot{f}\;)\;\;=\;\;\sum_{\alpha,\,\beta\,=\,1}^{n}\frac{\partial^{2}h^{\omega}}{\partial
f^{\alpha}\,\partial
f^{\beta}}(\,f\,)\;\delta\\!f^{\alpha}\;\dot{f}^{\beta}\;+\;\sum_{\alpha\,=\,1}^{n}\frac{\partial
h^{\omega}}{\partial f^{\alpha}}(\,f\,)\;\delta\\!\dot{f}^{\alpha}$

for the induced change of coordinates on $T(\,T\mathscr{F}\,)$ with similar
formulas for $\dot{h}^{\omega}$ and $\delta\\!h^{\omega}$. It is the symmetry
of $\frac{\partial^{2}h^{\omega}}{\partial f^{\alpha}\,\partial
f^{\beta}}(\,f\,)$ under the interchange $\alpha\,\leftrightarrow\,\beta$,
which allows $\Theta$ to read (23) in all local coordinates on
$T(\,T\mathscr{F}\,)$ induced by coordinates on $\mathscr{F}$! In taking
differences the quadratic term drops out, hence the fibers of $\Pi$ are affine
spaces modelled on $T\mathscr{F}$ under the difference map

(24)
$(\;f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\;\mathbb{F}^{\alpha}\;)\;-\;(\;f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\;\mathbb{\hat{F}}^{\alpha}\;)\;\;:=\;\;(\;f^{\alpha};\,\mathbb{F}^{\alpha}\,-\,\mathbb{\hat{F}}^{\alpha}\;)$

in all local coordinates on $T(\,T\mathscr{F}\,)$ induced by local coordinates
on $\mathscr{F}$.

Turning to fiber bundles over a manifold $M$ we choose local coordinates on
the total space of a fiber bundle by lifting local coordinates
$(\,x^{1},\,\ldots,\,x^{m}\,)$ of $M$ to functions on $\mathscr{F}M$ still
denoted by $x^{1},\,\ldots,\,x^{m}$. Complemented by suitable functions
$f^{1},\,\ldots,\,f^{n}\,\in\,C^{\infty}_{\mathrm{loc}}(\,\mathscr{F}M\,)$ we
obtain a system of local coordinates
$(\,x^{1},\,\ldots,\,x^{m};\,f^{1},\,\ldots,\,f^{n}\,)$ with the property that
the differentials $dx^{1},\,\ldots,\,dx^{m}$ of the coordinates constant along
the fibers of $\mathscr{F}M$ generate the horizontal forms so that the
vertical tangent bundle is spanned by:

$\mathrm{Vert}_{(\,x,\,f\,)}\mathscr{F}M\;\;:=\;\;\mathrm{span}_{\mathbb{R}}\Big{\\{}\;\frac{\partial}{\partial
f^{1}},\,\ldots,\,\frac{\partial}{\partial
f^{n}}\;\Big{\\}}\;\;\subseteq\;\;T_{(\,x,\,f\,)}\mathscr{F}M$

All the constructions discussed in the first part of this appendix for the
second iterated tangent bundle $T(\,T\mathscr{F}\,)$ extend without further
ado to the second iterated vertical tangent bundle
$\mathrm{Vert}\,\mathrm{Vert}\,\mathscr{F}M$ by throwing in the additional
coordinates
$(\,f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\,\delta\\!\dot{f}^{\alpha}\,)\,\rightsquigarrow\,(\,x^{\mu};\,f^{\alpha};\,\dot{f}^{\alpha};\,\delta\\!f^{\alpha};\,\delta\\!\dot{f}^{\alpha}\,)$
of the base manifold $M$ as passive coordinates. In this specific local
coordinate setup for a fiber bundle an arbitrary non–linear connection
$\mathbb{P}^{\nabla}$ reads

$\mathbb{P}^{\nabla}_{(\,x,f\,)}\Big{(}\frac{\partial}{\partial
f^{\alpha}}\Big{)}\;\;=\;\;\frac{\partial}{\partial
f^{\alpha}}\qquad\quad\mathbb{P}^{\nabla}_{(\,x,f\,)}\Big{(}\frac{\partial}{\partial
x^{\mu}}\Big{)}\;\;=\;\;\sum_{\alpha\,=\,1}^{n}\,\Gamma_{\mu}^{\alpha}(\,x,\,f\,)\;\frac{\partial}{\partial
f^{\alpha}}$

with suitable local coefficient functions $\Gamma^{\alpha}_{\mu}(\,x,\,f\,)$
generalizing the standard Christoffel symbols. Needless to say the curvature
$R^{\nabla}$ of the non–linear connection $\mathbb{P}^{\nabla}$ is determined
by these generalized Christoffel symbols, due to horizontality, it suffices to
calculate expression (4) for the vector fields:

$\displaystyle R^{\nabla}\Big{(}\,\frac{\partial}{\partial
x^{\mu}},\,\frac{\partial}{\partial x^{\nu}}\,\Big{)}$ $\displaystyle=$
$\displaystyle-\,\mathbb{P}^{\nabla}\left[\;\frac{\partial}{\partial
x^{\mu}}\;-\;\sum_{\alpha\,=\,1}^{n}\Gamma^{\alpha}_{\mu}(\,x,f\,)\,\frac{\partial}{\partial
f^{\alpha}},\;\frac{\partial}{\partial
x^{\nu}}\;-\;\sum_{\beta\,=\,1}^{n}\Gamma^{\beta}_{\nu}(\,x,f\,)\,\frac{\partial}{\partial
f^{\beta}}\;\right]$ $\displaystyle=$
$\displaystyle+\;\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\beta}_{\nu}}{\partial
x^{\mu}}(\,x,f\,)\,\frac{\partial}{\partial
f^{\beta}}\;-\;\sum_{\alpha,\,\beta\,=\,1}^{n}\Gamma^{\alpha}_{\mu}(\,x,f\,)\,\frac{\partial\Gamma^{\beta}_{\nu}}{\partial
f^{\alpha}}(\,x,f\,)\,\frac{\partial}{\partial f^{\beta}}$
$\displaystyle-\;\sum_{\alpha\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
x^{\nu}}(\,x,f\,)\,\frac{\partial}{\partial
f^{\alpha}}\;+\;\sum_{\beta,\,\alpha\,=\,1}^{n}\Gamma^{\beta}_{\nu}(\,x,f\,)\,\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}(\,x,f\,)\,\frac{\partial}{\partial f^{\alpha}}\ .$

Interchanging the dummy indices $\alpha\leftrightarrow\beta$ if necessary,
this results in a sum of the form $R^{\nabla}(\,\frac{\partial}{\partial
x^{\mu}},\,\frac{\partial}{\partial x^{\nu}}\,)\,=\,\sum
R^{\alpha}_{\mu\nu}(\,x,\,f\,)\,\frac{\partial}{\partial f^{\alpha}}$ with
coefficients given by:

(25)
$R^{\alpha}_{\mu\nu}\;\;:=\;\;\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
x^{\mu}}\;-\;\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
x^{\nu}}\;+\;\sum_{\beta\,=\,1}^{n}\Big{(}\;\Gamma^{\beta}_{\nu}\,\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}\;-\;\Gamma^{\beta}_{\mu}\,\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
f^{\beta}}\;\Big{)}$

In case the fiber bundle is actually a vector bundle we have a distinguished
class of local coordinates $(\,x^{1},\ldots,x^{m};\,v^{1},\ldots,v^{n}\,)$ on
the total space $VM$ associated to a basis
$\mathbf{v}_{1},\,\ldots,\,\mathbf{v}_{n}\,\in\,\Gamma_{\mathrm{loc}}(\,M,VM\,)$
of local sections:

$(\,x^{1},\,\ldots\,x^{m};\,v^{1},\,\ldots,\,v^{n}\,)\;\;\longleftrightarrow\;\;v^{1}\,\mathbf{v}_{1}\;+\;\ldots\;+\;v^{n}\,\mathbf{v}_{n}\;\;\in\;\;V_{(x^{1},\ldots,x^{m})}M\
.$

In these linear local coordinates on $VM$ the generalized Christoffel symbols
$\Gamma^{\alpha}_{\mu}(\,x,\,v\,)$ of a linear connection are linear functions
with an expansion

$\Gamma^{\alpha}_{\mu}(\,x^{1},\,\ldots,\,x^{m};\,v^{1},\,\ldots,\,v^{n}\,)\;\;=\;\;\sum_{\omega\,=\,1}^{n}\Gamma^{\alpha}_{\mu\omega}(\,x^{1},\,\ldots,\,x^{m}\,)\,v^{\omega}$

with local coefficient functions $\Gamma^{\alpha}_{\mu\omega}(\,x\,)$. In
particular the expansion (25) of the curvature of a linear connection
simplifies in linear coordinates on $VM$

$R^{\nabla}\Big{(}\;\frac{\partial}{\partial
x^{\mu}},\;\frac{\partial}{\partial
x^{\nu}}\;\Big{)}\;\;=\;\;\sum_{\alpha,\,\omega\,=\,1}^{n}R_{\mu\nu;\,\omega}^{\alpha}(\,x^{1},\,\ldots,\,x^{m}\,)\;v^{\omega}\;\frac{\partial}{\partial
v^{\alpha}}$

with coefficients $R_{\mu\nu;\,\omega}^{\alpha}(\,x\,)$ given by the classical
formula:

(26)
$R_{\mu\nu;\,\omega}^{\alpha}\;\;=\;\;\frac{\partial\Gamma_{\nu\omega}^{\alpha}}{\partial
x^{\mu}}\;-\;\frac{\partial\Gamma_{\mu\omega}^{\alpha}}{\partial
x^{\nu}}\;+\;\sum_{\beta\,=\,1}^{n}\Big{(}\;\Gamma_{\mu\beta}^{\alpha}\,\Gamma_{\nu\omega}^{\beta}\;-\;\Gamma_{\nu\beta}^{\alpha}\,\Gamma_{\mu\omega}^{\beta}\;\Big{)}\
.$

Coming back to fiber bundles we want to discuss the covariant derivative of a
local section $f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\mathscr{F}M\,)$ of
$\mathscr{F}M$, which in the chosen local coordinates on $M$ and
$\mathscr{F}M$ becomes an $n$–tuple of smooth functions:

$f(\,x^{1},\ldots,x^{m}\,)\;\;=\;\;(\,x^{1},\ldots,x^{m};\,f^{1}(x^{1},\ldots,x^{m}),\ldots,f^{n}(x^{1},\ldots,x^{m})\,)\
.$

In turn its covariant derivative with respect to the connection
$\mathbb{P}^{\nabla}$ reads:

$\displaystyle\Big{(}\,D^{\nabla}_{\frac{\partial}{\partial
x^{\mu}}}f\,\Big{)}(\,x\,)$ $\displaystyle=$
$\displaystyle\mathbb{P}^{\nabla}_{(\,x,f(x)\,)}\Big{(}\;\frac{\partial}{\partial
x^{\mu}}\;+\;\sum_{\alpha\,=\,1}^{n}\frac{\partial f^{\alpha}}{\partial
x^{\mu}}(\,x\,)\frac{\partial}{\partial f^{\alpha}}\;\Big{)}$ $\displaystyle=$
$\displaystyle\sum_{\alpha\,=\,1}^{n}\Big{(}\;\frac{\partial
f^{\alpha}}{\partial
x^{\mu}}(\,x\,)\;+\;\Gamma^{\alpha}_{\mu}(\,x,f(x)\,)\;\Big{)}\,\frac{\partial}{\partial
f^{\alpha}}\ ,$

In this formula we have considered the result as a vertical vector field,
alternatively we may write the result as a local section of
$\mathrm{Vert}\,\mathscr{F}M$

(27) $\Big{(}\,D^{\nabla}_{\frac{\partial}{\partial
x^{\mu}}}f\,\Big{)}(\,x\,)\;\;=\;\;\Big{(}\;x;\,f^{\alpha}(x);\,\frac{\partial
f^{\alpha}}{\partial
x^{\mu}}(x)\,+\,\Gamma^{\alpha}_{\mu}(\,x,\,f(x)\,)\;\Big{)}\ .$

For a smooth family $(\,f_{\varepsilon}\,)$ of local sections of
$\mathscr{F}M$ we obtain the variation

$\displaystyle\Big{(}{\textstyle\left.\frac{d}{d\varepsilon}\right|_{0}}D^{\nabla}_{\frac{\partial}{\partial
x^{\mu}}}f_{\varepsilon}\Big{)}(\,x\,)$ $\displaystyle=$
$\displaystyle\Big{(}\;x;\,f^{\alpha}(x);\,\frac{\partial f^{\alpha}}{\partial
x^{\mu}}(x)\,+\,\Gamma^{\alpha}_{\mu}(\,x,\,f(x)\,);$
$\displaystyle\delta\\!f^{\alpha}(x);\,\frac{\partial\,\delta\\!f^{\alpha}}{\partial
x^{\mu}}(x)+\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}(\,x,f(x)\,)\,\delta\\!f^{\beta}(x)\;\Big{)}$

with $f^{\alpha}\,:=\,f^{\alpha}_{0}$ and
$\delta\\!f^{\alpha}\,:=\,\left.\frac{d}{d\varepsilon}\right|_{0}f^{\alpha}_{\varepsilon}$.
Evidently we need to apply the fiberwise involution $\Theta$ to interpret the
result as the covariant derivative of the variation
$\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}\,\in\,\Gamma_{\mathrm{loc}}(\,M,\mathrm{Vert}\,\mathscr{F}M\,)$
with respect to the non–linear connection
$\mathbb{P}^{\nabla^{\mathrm{Vert}}}$ induced by equation (7) on the vertical
tangent bundle:

$\displaystyle\Big{(}\;D^{\nabla^{\mathrm{Vert}}}_{\frac{\partial}{\partial
x^{\mu}}}{\textstyle\left.\frac{d}{d\varepsilon}\right|_{0}f_{\varepsilon}}\;\Big{)}(\,x\,)$
$\displaystyle=$
$\displaystyle\Big{(}\;x;\,f^{\alpha}(x);\,\delta\\!f^{\alpha}(x);\,\frac{\partial
f^{\alpha}}{\partial x^{\mu}}(x)\,+\,\Gamma^{\alpha}_{\mu}(\,x,\,f(x)\,);$
$\displaystyle\qquad\frac{\partial\,\delta\\!f^{\alpha}}{\partial
x^{\mu}}(x)\,+\,\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}(\,x,f(x)\,)\,\delta\\!f^{\beta}(x)\;\Big{)}.$

Comparing the resulting expression with the formula (27) for a general
non–linear connection we conclude that the connection
$\mathbb{P}^{\nabla^{\mathrm{Vert}}}$ is given by

$\mathbb{P}^{\nabla^{\mathrm{Vert}}}_{(\,x,f,\delta\\!f\,)}\Big{(}\,\frac{\partial}{\partial
x^{\mu}}\,\Big{)}\;\;=\;\;\sum_{\alpha\,=\,1}^{n}\,\Gamma_{\mu}^{\alpha}(\,x,f\,)\;\frac{\partial}{\partial
f^{\alpha}}\;+\;\sum_{\alpha,\,\beta\,=\,1}^{n}\,\frac{\partial\Gamma_{\mu}^{\alpha}}{\partial
f^{\beta}}(\,x,f\,)\,\delta\\!f^{\beta}\,\frac{\partial}{\partial\,\delta\\!f^{\alpha}}\
,$

note that the vertical tangent vectors $\frac{\partial}{\partial f^{\alpha}}$
and $\frac{\partial}{\partial\,\delta\\!f^{\alpha}}$ are fixed by every
non–linear connection on $\mathrm{Vert}\,\mathscr{F}M$. Substituting the
variation $\delta\\!f^{\alpha}(x)$ by the corresponding expression
$\frac{\partial f^{\alpha}}{\partial
x^{\nu}}(x)\,+\,\Gamma^{\alpha}_{\nu}(x,f(x))$ in the covariant derivative
$D^{\nabla}_{\frac{\partial}{\partial x^{\nu}}}f$ of the local section
$f\,\in\,\Gamma_{\mathrm{loc}}(\,M,\mathscr{F}M\,)$ we obtain directly:

$\displaystyle\Big{(}\,D^{\nabla^{\mathrm{Vert}}}_{\frac{\partial}{\partial
x^{\mu}}}D^{\nabla^{\hphantom{V}}}_{\frac{\partial}{\partial
x^{\nu}}}f\,\Big{)}(\,x\,)$ $\displaystyle=$
$\displaystyle\Big{(}\;x;\,f^{\alpha}(x);\,\frac{\partial f^{\alpha}}{\partial
x^{\nu}}(x)\,+\,\Gamma^{\alpha}_{\nu}(\,x,f(x)\,);\,\frac{\partial
f^{\alpha}}{\partial x^{\mu}}(x)\,+\,\Gamma^{\alpha}_{\mu}(\,x,f(x)\,);$
$\displaystyle\;\;+\,\frac{\partial^{2}f^{\alpha}}{\partial x^{\mu}\partial
x^{\nu}}(x)\,+\,\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
x^{\mu}}(x)\,+\,\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}(\,x,f(x)\,)\,\Gamma^{\beta}_{\nu}(\,x,f(x)\,)$
$\displaystyle\;\;+\,\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
f^{\beta}}(\,x,f(x)\,)\,\frac{\partial f^{\beta}}{\partial
x^{\mu}}(x)\,+\,\sum_{\beta\,=\,1}^{n}\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}(\,x,f(x)\,)\,\frac{\partial f^{\beta}}{\partial
x^{\nu}}(x)\;\Big{)}\ .$

Evidently it is necessary to apply the involution $\Theta$ first to form the
difference

$\displaystyle\left(\;D^{\nabla^{\mathrm{Vert}}}_{\frac{\partial}{\partial
x^{\mu}}}D^{\nabla^{\hphantom{V}}}_{\frac{\partial}{\partial
x^{\nu}}}f\;-\;\Theta\Big{(}\,D^{\nabla^{\mathrm{Vert}}}_{\frac{\partial}{\partial
x^{\nu}}}D^{\nabla^{\hphantom{V}}}_{\frac{\partial}{\partial
x^{\mu}}}f\,\Big{)}\;\right)(\,x\,)$ $\displaystyle=$
$\displaystyle\Big{(}\;x;\,f^{\alpha}(x);\,\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
x^{\mu}}\,-\,\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
x^{\nu}}\,+\,\sum_{\beta\,=\,1}^{n}\Big{(}\,\Gamma^{\beta}_{\nu}\,\frac{\partial\Gamma^{\alpha}_{\mu}}{\partial
f^{\beta}}\,-\,\Gamma^{\beta}_{\mu}\,\frac{\partial\Gamma^{\alpha}_{\nu}}{\partial
f^{\beta}}\,\Big{)}\;\Big{)}$ $\displaystyle=$
$\displaystyle\Big{(}\;x;\,f^{\alpha}(x);\,R^{\alpha}_{\mu\nu}(\,x,\,f(x)\,)\;\Big{)}\;\;=\;\;\Big{(}\;R^{\nabla}_{\frac{\partial}{\partial
x^{\mu}},\,\frac{\partial}{\partial x^{\nu}}}f\;\Big{)}(\,x\,)\ ,$

where the coefficients $R^{\alpha}_{\mu\nu}(x,f)$ of the curvature of the
connection $\mathbb{P}^{\nabla}$ have been calculated in equation (25). Due to
$[\,\frac{\partial}{\partial x^{\mu}},\,\frac{\partial}{\partial
x^{\nu}}\,]\,=\,0$ the latter identity is exactly the local coordinates
formulation of the curvature identity (8).

## References

  * [1] Bleecker, D. : Gauge Theory and Variational Principles,  Global Analysis Pure and Applied, Addison–Wesley (1981).
  * [2] Kolár, I., Michor, P. W. & Slovák, J. : Natural Operations in Differential Geometry,  internet book http://www.mat.univie.ac.at/ michor/kmsbookh.pdf
  * [3] Saldaña, M, G, A. & Weingart, G. : Functoriality of Principal Bundles and Connections,  arXiv:1907.10231v2, 18 Apr 2020.


# Towards Using Social HRI for Improving Children’s Healthcare Experiences

Mary Ellen Foster
School of Computing Science
University of Glasgow
Glasgow, Scotland, United Kingdom
<EMAIL_ADDRESS>Ronald P. A. Petrick
Edinburgh Centre for Robotics
Heriot-Watt University
Edinburgh, Scotland, United Kingdom
<EMAIL_ADDRESS>

###### Abstract

This paper describes a new research project that aims to develop a social
robot designed to help children cope with painful and distressing medical
procedures in a clinical setting. While robots have previously been trialled
for this task, with promising initial results, the systems have tended to be
teleoperated, limiting their flexibility and robustness. This project will use
epistemic planning techniques as a core component for action selection in the
robot system, in order to generate plans that include physical, sensory, and
social actions for interacting with humans. The robot will operate in a task
environment where appropriate and safe interaction with children,
parents/caregivers, and healthcare professionals is required. In addition to
addressing the core technical challenge of building an autonomous social
robot, the project will incorporate co-design techniques involving all
participant groups, and the final robot system will be evaluated in a two-site
clinical trial.

## Introduction

Children regularly experience pain and distress in medical situations that can
produce both short-term (e.g., fear, distress, inability to perform
procedures) and long-term (e.g., needle phobia, anxiety) negative effects (?).
While a range of techniques have been demonstrated as effective methods for
managing such situations (e.g., breathing exercises, distraction techniques,
cognitive-behavioural interactions (?)), delivered through a variety of means
(e.g., distraction cards, kaleidoscopes, music, and virtual reality games),
recent studies have also explored the use of social robots as a tool for
managing child pain and distress during medical procedures (?; ?).

One significant drawback in many of these past studies is that the robots tend
to be teleoperated or use purely scripted behaviours with limited autonomy and
responsiveness, thereby reducing the overall flexibility and robustness of the
system. Such approaches also limit the ability to personalise or adapt the
system to different user groups (e.g., children, parents of children
undergoing medical procedures, and medical practitioners), in order to provide
appropriate context-dependent behaviours and responses.

Figure 1: A child interacting with the Nao robot.

From a technical point of view, a fundamental component in any social robot is
the action selection system which controls overall robot behaviour: the robot
must make high-level decisions as to which spoken, non-verbal, and task-based
actions should be taken next by the system as a whole. It is also crucial not
only to choose the appropriate action, but also to monitor the execution of
such actions and the state of the world detected by the robot’s sensors:
particularly in the context of embodied interactions with a robot, it is
likely that the predicted state will often differ from the sensed world state,
due to both the unexpected behaviour of the human interaction partners as well
as the inherent uncertainty involved in sensing and acting in the physical
world.

This paper describes a new project (?) aimed at addressing this limitation by
developing and evaluating a clinically relevant and responsive social robot,
using the Nao robot platform from SoftBank Robotics (Figure 1). While the
majority of social robotics systems generally use either scripted behaviour
for action selection, or machine learning approaches to learn the correct
responses to user actions given sample inputs, we will instead use epistemic
planning techniques (?; ?) as a key aspect of our approach, as the basis for
high-level action selection and execution monitoring (?). In the remainder of
this paper we outline the relevant related work, proposed technical
architecture, and challenges on this project.

## Related Work

The area of socially assistive robotics (?) generally focuses on designing a
robot system with the goal of creating an effective interaction with a human
partner for the purpose of providing assistance and achieving measurable
progress in a defined domain. Such robots have been used to improve the
cognitive abilities of adult Alzheimer’s patients (?), to alleviate feelings
of loneliness and depression in the elderly (?), and to help adults with
autism to improve work-related social skills (?). An important application of
socially assistive robots also focuses on autism in children, where robots
have been used for diagnosis, intervention, and therapy (?). Since children
often perceive social robots as being similar to a companion animal or pet,
such robots have also been used for play therapy and social learning (?).

Our project will explore the use of socially assistive robots for reducing
child distress and pain in clinical settings. (?) recently examined studies
where a robot was used in this context: overall, while the results seem
promising and suggest that the robots succeeded in reducing pain, a need for
improved methodology and measures was identified. In particular, the authors
suggest more effective approaches could be created by ensuring healthcare
experts and system engineers collaborate from the start, and that user and
family partners contribute to a user-centred design process. Our planned work
includes input from all such groups as part of the research team (see below).

On the technical side, planning and interaction have a long history, and
planning techniques have been applied previously in a range of social robots
and interactive systems—recent examples include (?; ?; ?; ?). The most similar
approach to ours is the JAMES social robot bartender (?; ?), which directly
used an automated planner to choose the robot’s physical, sensing, and
interactive actions. This system will form the basis of the approach used on
this project. Recent work on explainable planning (?) has also highlighted the
links between planning and user interaction, and is relevant to this work.

## Robot Behaviour and Task Environment

The high-level goal of the project is to develop and evaluate an autonomous
social robot designed to help children deal with procedural pain in emergency
room environments. The behaviour of the robot will be based on existing
cognitive behavioural interventions that have been demonstrated to be
effective in this context (e.g., distraction, empathy). The target robot
platform is the Nao robot from SoftBank Robotics (Figure 1), which has been
widely used in child-robot interaction studies, including in the identical
clinical context we are targeting (?). In addition to lab-based testing with
the robot, the system will be tested in the target environment throughout the
project period, culminating in a two-site randomised clinical trial at the end
of the project.

At the action level, the robot will engage in physical, sensory, and social
actions. Physical actions will include movement of the robot as a whole
(movement between locations, using build-in features like dancing) and moving
various aspects of the embodiment (e.g., arm and head movements). Sensory
actions will include targeted information gathering through available sensor
modalities (vision, speech) about the social context and task, but also
through verbal interaction (question asking). Social actions will include a
range of dialogue and interactive actions (question answering, engagement
behaviour, explanation). Moreover, the execution of some actions may change
depending on the human user involved and the social context (e.g., answering a
question for a parent versus a medical professional, interacting with a happy
child versus one who is crying). The following robot utterances have been used
in previous studies (?; ?) and are also relevant here:

  * •

“I am excited to play with you today.”

  * •

“I need to catch my breath and control my breathing. Let’s try this together!
Breathe in through your nose, hold your breath for two seconds, and out
through your mouth. Try it with me!”

  * •

“I understand you are in a lot of pain right now and I’m here to help you.”

  * •

“I think we should celebrate how brave you are. Here’s a new dance I just
learned.”

While the robot hardware platform itself supports a range of built-in
features, which can be abstracted into high-level behaviours to support the
needed actions, the actual decision as to which behaviours are included will
be decided as part of a co-design process that involves the participation of
children, parents, healthcare professionals, as well as the technical research
team. This process therefore has implications for the technical aspects of the
system design, including the process of modelling such actions for use with
the action selection and decision-making components of the system.

## Overview of the Robot System

The robot system will include components for social signal processing, high-
level behaviour selection, and execution monitoring and recovery as shown in
Figure 2. The system will be based on the JAMES robot bartender system (?),
which uses a knowledge-level planner to generate plans for social interaction.

Figure 2: Software architecture for the robot system.

Social signal recognition: A core task in the system is to use the information
from the robot’s built-in audio-visual sensors (possibly combined with
environmental sensors) to determine the state of people (children, adults) in
the task area. The particular states to be detected will be determined through
a combination of the capabilities of the sensors, as well as the states
determined to be relevant from the design process. Based on the detected
verbal and non-verbal cues employed by all humans near the robot, the state
will be estimated, including the physical state and social aspects such as
attitudes, emotions, and intention. We plan to use a neural-network approach
to detect the states, similar to the approach of (?).

Behaviour selection: Using the available robot actions and the detected social
states from the social signal recogniser, the system will choose appropriate
high-level actions to be performed by the robot. Action selection is performed
using PKS (Planning with Knowledge and Sensing) (?; ?), an epistemic planner
that operates at the knowledge level and reasons about how its knowledge
state, rather than the world state, changes due to action. PKS has been
previously used in interactive environments, such as the JAMES robot
bartender, where information-gathering dialogue actions were modelled as
sensing actions in the planner (?).

Social signal generation: Once a behaviour is selected, the social signal
generation process generation converts these high-level actions into concrete
robot-level actions that can be executed on the Nao robot platform. As output,
this component will produce multimodal behaviour plans, including verbal and
non-verbal actions, that are coordinated temporally and spatially for robot
execution.

Execution monitoring and recovery: The system will monitor the changes to the
state (as detected by the social signal recogniser) while planned actions are
executed on the robot, using traditional plan monitoring techniques, but
involving the richer social and epistemic states afforded by the state
recognition and planning processes. Due to the inherent uncertainty of robot
sensors and the unpredictable behaviour of humans it is likely that the
planned states will often differ from the actual world state. The monitoring
system will detect such mismatches and determine whether the execution of the
current high-level plan should continue or whether a new plan is needed,
invoking replanning as needed.

All software components will be developed using the Robot Operating System
(ROS) as the core middleware, together with ROSPlan (?) framework for
integrating the planning tools.

## Challenges

From a high-level planning perspective, the task of applying planning in a
child-centred medical context centres around an important knowledge
engineering task to accurately modelling the required states, actions, and
goals that reflect the types of activities the robot is expected to perform.
The high-level planner is responsible for selecting robot actions to respond
appropriately in the current social state of the task, with a mix of physical,
sensory, and social behaviours. However, unlike many robot systems which are
designed from the ground up by the technical/research team, this project
involves a wider collaboration at the design phase. Furthermore, the output of
the robot may have to be tailored to different user groups, depending on the
interaction situation. Finally, the entire robot system needs a high degree of
robustness since the user studies will include a full clinical trial. We
highlight some of these challenges below.

Co-design of robot capabilities: At a high level, the robot must select
appropriate actions to support children undergoing clinical procedures.
However, the details of the exact behaviours and features will be developed
through a co-design approach using the principles of user-centred interaction
design (?). The perspectives of all groups involved in the task will be
considered, including children, parents/caregivers, healthcare professionals,
as well as the research and technical team. This process will consider all
aspects of the task including the needs of children, caregivers, and the
medical team during a clinical procedure, the (possibly differing) perceptions
these groups have of the robot, and the core functionality required for the
task. While this process will be somewhat constrained by the physical
limitations of the robot platform, and the representational and reasoning
capabilities of the planner, the knowledge engineering task of building
planning models cannot be fully realised without the input from the co-design
phase.

Personalised interaction: The robot system is meant to interact with different
user groups, from children to healthcare professionals. The kinds of plans
that are generated for the various groups will also differ, as will the
potential execution of those plans on the robot. For instance, answering a
child’s question may be realised differently compared with a similar question
from a parent or healthcare professional. Some of these differences will be
reflected at the action level in the planning domain model, which also
introduces some interesting connections to explainable planning (XAIP) (?). In
particular, explainability of planning decisions and plan execution will be
explored on the project, especially in the context of analysing planner
decisions by healthcare professionals.

System robustness and safety: Several user studies are planned for the
project, from initial lab-based testing to a clinical trial, to understand the
behaviour of the system. The end system is meant to be run in a real clinical
setting (at least for evaluation purposes), with minimal human intervention.
Moreover, the system is meant to run in the presence of different user groups,
including both children and adults. As a result, the system will require a
high degree of safety and robustness at all levels. In particular, recognising
children’s social signals in a real-world setting is expected to present a
particular technical challenge, and we will explore a range of sensor
configurations and audiovisual processing approaches to find the one that
works best in this specific domain. In the final phase of the project, the
clinical trial will evaluate the primary goal of the project: that interaction
with a robust, adaptive, socially intelligent robot can effectively support
children during a clinical procedure and reduce their distress and pain. The
clinical trial is planned for the two Canadian paediatric departments where
the co-design and usability studies will also take place.

## Summary and Conclusions

This paper describes a new research project that plans to design and use a
social robot in a medical setting to support children undergoing painful
clinical procedures. At the core of the action selection and decision-making
system will be a high-level epistemic planner that will be responsible for
generating plans to interact with different user groups, including children,
parents/caregivers, and healthcare professionals. The robot behaviour will
include physical robot actions, sensing actions, and social actions for
guiding the interaction. A co-design process will be used, involving all the
user groups along with the research/technical team, which will feed into the
design of the planning model. Some aspects of explainable planning and plan
personalisation will also be necessary on the project. The system will be
demonstrated and evaluated through a series of usability studies, leading up
to a full clinical trial in a real-world medical setting. The hope is that
this work will extend existing work on social robotics and demonstrate the
potential of automated planning as a useful tool for autonomous decision
making in this challenging domain.

## Acknowledgements

This work is funded by the ESRC/SSHRC Canada-UK Artificial Intelligence
Initiative through grant ES/T012986/1. We also acknowledge the lead Canadian
research partners on the project: Samina Ali, Sasha Litwin, Jennifer Parker,
David Harris Smith, Jennifer Stinson, and Frauke Zeller.

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