# Group-Theoretic Matching of the Length and Equality Principles in Geometry

Serhii Samokhvalov, Olena Balakireva  Dniprovsk State Technical University,
Kamianske, UKRAINE<EMAIL_ADDRESS>
###### Abstract

Deformed generalized gauge groups, which were created from physical
considerations and made it possible to clarify some long-standing problems in
physics, such as the problem of motion and the problem of the energy of the
gravitational field, simultaneously carried out the implementation of the
Klein’s Erlangen program for spaces with variable curvature, and for
Riemannian spaces even in two different ways. In this paper, this issue is
considered from a geometric point of view and these two methods of group-
theoretical description of Riemannian spaces are reconciled. The paper deals
with the canonical deformed group of diffeomorphisms with a given length scale
which describes the motion of unit scales in a Riemannian space. This allows
one to measure the lengths of arbitrary curves implementing the length
principle which was laid by B. Riemann at the foundation of geometry. We
present a method of univocal extension of this group to a group which contains
gauge rotations of vectors (the group of parallel transports) whose
transformations leave unchanged the lengths of vectors and the corners between
them, implementing for Riemannian spaces the Klein’s principle of equality and
matching both principles of the foundations of geometry, thus overcoming the
Riemann-Klein antagonism.

Keywords: deformed group of diffeomorphisms, Klein’s Erlangen Program,
Riemannian space, Riemann-Klein antagonism

## Introduction

Creation of Lobachevskii geometry and Gauss’ works on the theory of surfaces
stimulated the intensive development of geometric theories in the 19-th
century. As a result, the works of B. Riemann [1] and F. Klein [2] were
appeared. These works laid different principles at the foundation of the
geometry: the length principle which requires the possibility to measure the
lengths of arbitrary lines no matter how they are situated, and equality
principle which is established by coincidence of figures in the space by means
of transformations belonging to a group of transformations of the space – the
principal group of the geometry under consideration (according to F. Klein).
According to E. Cartan [3], there is an antagonism between these two
principles owning to the absence of any homogeneity in an arbitrary curved
Riemannian space.

Earlier, attempts were made to overcome this antagonism by means of refusing
from group structure of used transformations. E. Cartan suggested to consider
a curved space as a anholonomic space with the same principal (fundamental
according to the terminology used by Cartan) group as the corresponding flat
space [3]. R. Sulanke and P. Wintgen [4] applied the category theory for
description of curved spaces, and L. V. Sabinin [5] used quasigroups replying
on the thesis that nonassociativity is an algebraic equivalent of the
geometric notion of curvature.

It has been clarified later that so-called deformed generalized gauge groups
introduced from physical arguments [6] can describe, and in two different
ways, geometric structure of variable curvature [7], in particular, affinely
connected spaces [8] and Riemannian spaces [9].

In the first case, the translation generators of a group coincide with
covariant derivatives in a curved affinely connected space. The curvature and
the torsion of the space are determined by the structure functions of the
group, which are antisymmetric parts of the coefficients of the second order
terms in the expansion of the multiplication law of the group with respect to
parameters [8].

In the second case, it has become possible to use for torsion-free affinely
connected spaces, in particular, for Riemannian spaces, a more narrow group,
namely, the deformed group of diffeomorphisms. Acting on a manifold, it
generates an action on its tangent bundle and thus defines a rule of parallel
translations of vectors. But, in this case, for description of the curvature
tensor, one needs to know the third terms of the expansion of the
multiplication law of the group with respect to parameters [9].

In this paper, we suggest a method which makes it possible to match and unite
these two approaches to group-theoretical description of Riemannian spaces and
thus to overcome the Riemann-Klein antagonism.

The method we use eliminates the indefiniteness which took place in previous
papers [8],[9] with respect to finite translations of vectors given in curved
spaces by action of deformed generalized gauge groups. This has been achieved
by means of a generalization to the case under consideration of the notion of
canonical Lie groups, which establishes one-to-one correspondence between the
groups and the geometric structures defined by them. We present criteria of
canonicity of generalized gauge groups under consideration.

We show that a canonical deformed group of diffeomorphisms with given length
scale – the group of Riemannian translations $RT$ – makes it possible to
measure the lengths of geodesics by means of translation of the unit scale
along itself, which realizes the length principle (Riemann’s approach). What
is more, the group $RT$ contains in itself all characteristics of Riemannian
space, in particular, the curvature as a characteristic of deviation of
geodesics, and defines by its action on a manifold a Riemannian structure (by
means of geometric objects of a Riemannian geometry such as vector and
orthonormal frame fields, and, therefore, the metric). But the group $RT$ does
not preserve invariants on a set, and, therefore, cannot be considered as the
principal group of a Riemannian space.

However, as is shown in this paper, a given Riemannian structure on a manifold
(a metric) can be defined by different groups $RT$ which are related to each
other by nonlocal automorphisms of special form. The set of such automorphisms
is a group which is, in the case the requirement of its canonicity is imposed,
the group $DP$ of parallel translations of vectors in a Riemannian space. The
group $DP$, acting in the tangent bundle of a manifold, preserves the lengths
of vectors and the angles between vectors. Thus, the group of parallel
transports $DP$ can be considered as the principal group of a Riemannian space
because it does not change its characteristics and enables ones to establish
the equality (congruence) of vectors and their mutual disposition even in the
case they are given in different points (Klein’s approach). Here the curvature
plays the role of noncommutativity of covariant derivatives – the generators
of translations of $DP$ (or of the rotations of a vector when it is displaced
along a closed contour). Acting in the bundle of orthonormal frames on a
manifold (a principal bundle), the group $DP$ describes motions of frames,
which is an interpretation of E. Cartan’s method of moving frames [10], though
the structure equation of s Riemannian space in this case is not postulated as
it is within Cartan’s approach, but is a consequence of the existence of a
group $DP$.

The group of parallel transports $DP$, which realizes for Riemannian space the
equality principle, has as a subgroup the group of Riemannian translations
$RT$, which realizes the length principle and, therefore, unites the two
approaches laid at the foundation of geometry by B. Riemann and F. Klein, thus
overcoming the Riemann-Klein antagonism.

In the paper, we do not consider global-topological questions, all relations
are obtained within a coordinate domain of manifold, though we use more
adopted in contemporary geometry approach which avoids the use of coordinates.
In addition, by groups we mean the corresponding local groups, and some
terminology and notations concerning Lie groups follows [11].

##  1 Deformed gauge group of translations

The gauge group of translations $T_{X}^{g}$ of an affine space
$(X,\widetilde{T})$ whose elements are vector functions $\widetilde{t}_{x}$
with values in the vector space $\widetilde{T}$ which depend smoothly on
$x\subset X$ and satisfy the condition $|1+\partial_{x}\widetilde{t}_{x}|\neq
0$, where $\partial_{x}:=\partial/\partial x$, has the following
multiplication law:

$(\widetilde{t}\times\widetilde{t^{\prime}})_{x}=\widetilde{t}_{x}+\widetilde{t^{\prime}}_{x^{\prime}},\quad
x^{\prime}=x+\widetilde{t}_{x},\quad\widetilde{t}_{x},\,\widetilde{t^{\prime}}_{x^{\prime}}\subset
T_{X}^{g}.$ (1)

Thus, $T_{X}^{g}$ is group of diffeomorphisms of the space $X$ in additive
parameterization with parameters $\widetilde{t}_{x}$.

We see that the multiplication law in the group $T_{X}^{g}$ gives a rule for
the addition of vectors specified at different points, which is a
characteristic feature of generalized gauge groups, which is manifested in
their important role in describing geometric structures in which parallel
transport takes place.

The subset of elements of the group $T_{X}^{g}$, which is parameterized by
vectors independent of $x$, obviously forms the group $\widetilde{T}$.

Deformed gauge translation group $T_{X}^{gH}$ (deformed group of
diffeomorphisms) is by definition isomorphic to the group $T_{X}^{g}$ and is
parametrized by vector functions $t_{x}=H_{x}(\widetilde{t}_{x})$ taking
values in a vector space $T$ isomorphic to $\widetilde{T}$, where
$H_{x}:\widetilde{T}\rightarrow T$ is an invertible smoothly depending on
$x,\>\forall x\in X$ a deformation map, $K_{x}:=H_{x}^{-1}$. The mapping
$H_{x}$ is generally nonlinear and, therefore, is not an isomorphism between
the vector spaces $\widetilde{T}$ and $T$.

The multiplication law and the action of the group $T_{X}^{gH}$ on $X$ are
determined by its isomorphism to the group $T_{X}^{g}$ [9]:

$\eqalign{{(t\times
t^{\prime})_{x}=\varphi_{x}(t_{x},t^{\prime}_{x^{\prime}}):=H_{x}(K_{x}(t_{x})+K_{x^{\prime}}(t^{\prime}_{x^{\prime}})),}\cr{x^{\prime}=f_{x}(t_{x}):=x+K_{x}(t_{x}).}}$
(2)

The associativity law of multiplication and the composition law of the action
of the group $T^{gH}_{X}$ on $X$ are expressed by the following equations for
the mappings $\varphi_{x}$ and $f_{x}$:

$\varphi_{x}(\varphi_{x}(t,t^{\prime}),t^{\prime\prime})=\varphi_{x}(t,\varphi_{x^{\prime}}(t^{\prime},t^{\prime\prime})),\quad
f_{x^{\prime}}(t^{\prime})=f_{x}(\varphi_{x}(t,t^{\prime})),$ (3)

which are fulfilled due to the deformation method of constructing the group
$T^{gH}_{X}$ and whose solution is equalities (2) with an arbitrary
deformation mapping $H_{x}$ and constant parameters. The subset of elements of
the deformed group of diffeomorphisms $T^{gH}_{X}$, parametrized by vectors
independent of $x$, for an arbitrary deformation, in contrast to the
undeformed case, is not closed with respect to the multiplication operation,
and therefore no longer forms a group.

A consequence of equations (3) are equations that generalize the Lie equations
of the theory of Lie groups for the group $T^{gH}_{X}$:

$\mu_{x}(t)\cdot\partial\varphi_{x}(t,t^{\prime})=\mu_{x}(\varphi_{x}(t,t^{\prime}))+e_{x}\varphi_{x}(t,t^{\prime}),$
(4)
$\lambda_{x^{\prime}}(t^{\prime})\cdot\partial^{\prime}\varphi_{x}(t,t^{\prime})=\lambda_{x}(\varphi_{x}(t,t^{\prime})),$
(5) $\mu_{x}(t)\cdot\partial
f_{x}(t)=e_{x}f_{x}(t),\quad\lambda_{x}(t)\cdot\partial
f_{x}(t)=k_{x^{\prime}},$ (6)

where $\mu_{x}(t^{\prime}):=\partial\varphi_{x}(0,t^{\prime})$,
$\lambda_{x}(t):=\partial^{\prime}\varphi_{x}(t,0)$ are auxiliary maps,
$e_{x}:=k_{x}\cdot\partial_{x}$ are the generators of the action of the group
$T^{gH}_{X}$ on $X$, $k_{x}:=\partial K_{x}(0)$ are the deformation
coefficients (here it is assumed $\partial:=\partial/\partial t$,
$\partial^{\prime}:=\partial/\partial t^{\prime}$).

The integrability conditions for equations (4) - (6) are the equations that
generalize for the group $T^{gH}_{X}$ the Maurer – Cartan equations of the
theory of Lie groups:

$e_{x}\mu_{x}(t)\langle
l,l^{\prime}\rangle+\partial\mu_{x}(t)\langle\mu_{x}(t)\langle
l^{\prime}\rangle,l\rangle-(l\leftrightarrow l^{\prime})=\mu_{x}(t)\langle
C_{x}\langle l,l^{\prime}\rangle\rangle,$ (7)
$\partial\lambda_{x}(t)\langle\lambda_{x}(t)\langle
l\rangle,l^{\prime}\rangle-(l\leftrightarrow l^{\prime})=\lambda_{x}(t)\langle
C_{x^{\prime}}\langle l,l^{\prime}\rangle\rangle,$ (8) $e_{x}k_{x}\langle
l,l^{\prime}\rangle-(l\leftrightarrow l^{\prime})=k_{x}\langle C_{x}\langle
l,l^{\prime}\rangle\rangle,$ (9)

where $l,l^{\prime}\in T$ and

$C_{x}\langle l,l^{\prime}\rangle:=\gamma_{x}\langle
l,l^{\prime}\rangle-(l\leftrightarrow l^{\prime})$ (10)

is the structure operator of the group $T^{gH}_{X}$ that depends explicitly on
$x$ (in contrast to the case of finite-dimensional Lie groups), which is the
antisymmetric part of the mapping
$\gamma_{x}:=\partial\partial^{\prime}\varphi_{x}(0,0)$, which defines the
second order of the group $T^{gH}_{X}$ multiplication law and depends
explicitly on $x$ due to the explicit the dependence of the mapping
$\varphi_{x}$ of the group $T^{gH}_{X}$ on $x$.

The integrability condition for equations (7) - (9) is the Jacobi relation for
the structure operator, which in our case takes the form

$e_{x}C_{x}\langle l,l^{\prime},l^{\prime\prime}\rangle+C_{x}\langle
l,C_{x}\langle
l^{\prime},l^{\prime\prime}\rangle\rangle+cycl(l,l^{\prime},l^{\prime\prime})=0.$
(11)

Differentiation of equation (7) with respect to $t$ at zero allows us to
reveal an important relation

$e_{x}\gamma_{x}\langle l,l^{\prime},t\rangle+\gamma_{x}\langle
l,\gamma_{x}\langle l^{\prime},t\rangle\rangle-(l\leftrightarrow
l^{\prime})=R_{x}\langle t,l,l^{\prime}\rangle+\gamma_{x}\langle C_{x}\langle
l,l^{\prime}\rangle,t\rangle$ (12)

for the antisymmetric part

$R_{x}\langle t,l^{\prime},l\rangle:=\rho_{x}\langle
l,l^{\prime},t\rangle-(l\leftrightarrow l^{\prime})$ (13)

of the mapping $\rho_{x}:=\partial\partial^{\prime 2}\varphi_{x}(0,0)$, which
partially determines the third order of the multiplication law for the group
$T^{gH}_{X}$. The mapping $R_{x}$ will be called the curvature operator of the
group $T^{gH}_{X}$.

Definition (13) and the symmetry of the mapping $\rho_{x}$ in the last two
arguments imply the cyclic identity

$R_{x}\langle
l,l^{\prime},l^{\prime\prime}\rangle+cycl(l,l^{\prime},l^{\prime\prime})=0,$
(14)

which, due to Eq. (12), reduces to the Jacobi relation (11).

From a geometric point of view, the generators $e_{x}$ of the group
$T^{gH}_{X}$ define a field of affine frames on $X$, and its structure
operator $C_{x}$, due to Eq. (9), is its anholonomic object.

The multiplication law for the group $T^{gH}_{X}$, written for the
infinitesimal parameters
$(t\times\theta)_{x}=t_{x}+\theta_{x^{\prime}}+\gamma_{x}\langle
t_{x},\theta_{x^{\prime}}\rangle$, gives a rule for the composition of the
vectors $t_{x}$ and $\theta_{x^{\prime}}$ given at different points $x$ and
$x^{\prime}$, and therefore and a certain rule for parallel transport of the
vector $\theta_{x^{\prime}}$ from point $x^{\prime}$ to point $x$:

$\theta_{x||}:=\theta_{x^{\prime}}+\gamma_{x}\langle
t_{x},\theta_{x^{\prime}}\rangle=\theta_{x}+t_{x}\cdot\nabla_{x}\theta_{x},$
(15)

with which the composition law of the group $T^{gH}_{X}$ describes the
addition of vectors given already at one point:
$(t\times\theta)_{x}=t_{x}+\theta_{x||}$. Here $\nabla_{x}=e_{x}+\gamma_{x}$
is the covariant derivative, and in this sense the mapping $\gamma_{x}$ plays
the role of an object of affine connection in the frame $e_{x}$. Due to Eq.
(10), there is no torsion in our case.

The curvature operator $R_{x}$ of the group $T^{gH}_{X}$, by virtue of Eq.
(12), which in this case plays the role of the structure equation of the
affinely connected space, is its curvature tensor.

Using (2), we find

$\gamma_{x}\langle l,l^{\prime}\rangle=h_{x}\langle\Gamma_{x}\langle
k_{x}\langle l\rangle,k_{x}\langle l^{\prime}\rangle\rangle+e_{x}k_{x}\langle
l,l^{\prime}\rangle\rangle,$ (16)

where $h_{x}:=k^{-1}_{x}$ and
$\Gamma_{x}:=k_{x}\circ\tilde{\partial}^{2}H_{x}(0)$, here
$\tilde{\partial}:=\partial/\partial\tilde{t}$. Thus, the covariant derivative
can be represented as $\nabla_{x}=h_{x}\circ k_{x}\cdot\tilde{\nabla}_{x}\circ
k_{x}$, where $\tilde{\nabla}_{x}=\partial_{x}+\Gamma_{x}$ is the covariant
derivative in the frame $\partial_{x}$. Consequently, a smooth symmetric
bilinear map $\Gamma_{x}$ is an object of torsion-free affine connection in
the holonomic frame $\partial_{x}$, and since it is other-wise arbitrary due
to the arbitrariness of the deformation mappings $H_{x}$, the second order of
which is given by $\Gamma_{x}$, an arbitrary torsion-free affine connection
can be described in this way.

Theorem 1. Acting on a manifold $X$, the deformed gauge group of translations
$T^{gH}_{X}$ defines on it the structure of an affinely connected torsion-free
space whose structure equation is a necessary condition for the existence of
the group $T^{gH}_{X}$. An arbitrary torsion-free affine connection on $X$ can
be defined in this way.

Obviously, the group $T^{g}_{X}$ defines on $X$ the structure of a flat affine
space.

##  2 Canonical deformations

If in the expansion of the deformation map

$H_{x}(\tilde{t})=h_{x}\circ(\tilde{t}+\frac{1}{2}\Gamma_{x}\langle\tilde{t},\tilde{t}\rangle+\ldots),$

with the help of which the group $T^{gH}_{X}$ is constructed, the first order
determines the field of affine frames on $X$, and the second - the affine
connection, then all other orders do not affect to the connection defining by
the group $T^{gH}_{X}$ and can be arbitrary. To remove this ambiguity, we
generalize the notion of canonicity of Lie groups for the groups $T^{gH}_{X}$.

A Lie group $G$, $g\in G$, is canonical if any line of the form $g(s)=s\tau$
is its one-parameter subgroup [11]. In this case, the mapping
$\varphi(g,g^{\prime}):=g\times g^{\prime}$, which defines the multiplication
law in the group $G$, has the property
$\varphi(s\tau,s^{\prime}\tau)=(s+s^{\prime})\tau$ and is uniquely
reconstructed from the structure operator, and hence the Lie algebra of the
group $G$.

Definition 1. A deformed gauge translation group $T^{gH}_{X}$, as well as the
deformation with which it was obtained, will be called canonical if, for any
two points $x,x^{\prime}\in X$ there exists a smooth parametric curve
$x^{\prime}=x(s)$ such that

$(s\tau\times
s^{\prime}\tau)_{x}=\varphi_{x}(s\tau_{x},s^{\prime}\tau_{x^{\prime}})=(s+s^{\prime})\tau_{x},$
(17)

where
$\tau_{x^{\prime}}=h_{x^{\prime}}\langle\tilde{\tau}_{x^{\prime}}\rangle$ and
$\tilde{\tau}_{x^{\prime}}=\dot{x}^{\prime}$.

Differentiating definition (17) with respect to $s$ and $s^{\prime}$ at zero
simultaneously, we obtain the equation of the given curve
$\dot{\tau}_{x}+\gamma_{x}\langle\tau_{x},\tau_{x}\rangle=0$, or taking into
account (16) $\ddot{x}+\Gamma_{x}\langle\dot{x},\dot{x}\rangle=0$. Thus, for
an arbitrary torsion-free space with affine connection, the structure of which
is given by the action of the group $T^{gH}_{X}$, such a curve exists and is a
geodesic in the affine parameterization in the space of affine connection
defined by the group $T^{gH}_{X}$.

Criteria for canonicity. Along the geodesic connecting two arbitrary points
$x,x^{\prime}\in X$, the auxiliary functions of the canonical group
$T^{gH}_{X}$ satisfy the equations

$\mu_{x}(s\tau_{x})\langle\tau_{x}\rangle=\tau_{x}+s\gamma_{x}\langle\tau_{x},\tau_{x}\rangle,$
(18) $\lambda_{x}(s\tau_{x})\langle\tau_{x^{\prime}}\rangle=\tau_{x},$ (19)

Equation (18) is obtained by differentiating definition (17) with respect to
$s$, and (19), with respect to $s^{\prime}$ at zero.

Taking into account the equality $\rho_{x}\langle
l^{\prime},l,l\rangle=\partial^{2}\mu_{x}(0)\langle l,l,l^{\prime}\rangle$ and
twice differentiating Eq. (18) with respect to $s$ at zero, we obtain
$\rho_{x}\langle
l,l^{\prime},l^{\prime\prime}\rangle+cycl(l,l^{\prime},l^{\prime\prime})=0$,
which, using identity (14) and definition (13), allows us to establish the
following relation $\rho_{x}\langle
l^{\prime},l,l\rangle=\frac{2}{3}R_{x}\langle l,l,l^{\prime}\rangle$, which
holds for the canonical groups $T^{gH}_{X}$.

By means of deformation mappings, the auxiliary mapping $\lambda_{x}$, based
on the first formula in (2), is defined as follows:

$\lambda_{x}(t)=\tilde{\partial}H_{x}(\tilde{t})|_{\tilde{t}=K_{x}(t)}\circ
k_{x^{\prime}},$ (20)

as a result of which criterion (19) can be represented in the form
$\tilde{\partial}H_{x}(\tilde{t}_{x})\langle\tilde{\tau}_{x^{\prime}}\rangle=\dot{H}_{x}(\tilde{t}_{x})=\tau_{x}$,
whence it follows

Proposition 1. Along the geodesic connecting arbitrary two points separated by
a finite interval $\tilde{t}_{x}=x(s)-x(0)$, the corresponding parameter
$t_{x}=H_{x}(\tilde{t}_{x})$ of the canonical group $T^{gH}_{X}$ is
proportional to the initial vector of the geodesic: $t_{x}=s\tau_{x}$.

Thus, in the case of a canonical deformation, the function
$x^{\prime}=x+K_{x}(s\tau_{x})$ is a solution to the geodesic equation with
initial data $x$, $\dot{x}=k_{x}\langle\tau_{x}\rangle$. The parameters
$t_{x}$ of the canonical group $T^{gH}_{X}$ are elements of the tangent bundle
$TX$ and are similar to geodesic Riemannian coordinates outgoing from the
point $x$.

By virtue of criterion (19), the functions

$u(x^{\prime},\tilde{\tau}_{x^{\prime}}):=\partial_{x^{\prime}}H_{x}(x^{\prime}-x)\langle\tilde{\tau}_{x^{\prime}}\rangle$
(21)

are the first integrals of the geodesic equations
$\dot{x}^{\prime}=\tilde{\tau}_{x^{\prime}}$,
$\dot{\tilde{\tau}}_{x^{\prime}}=-\Gamma_{x^{\prime}}\langle\tilde{\tau}_{x^{\prime}},\tilde{\tau}_{x^{\prime}}\rangle$
and, therefore, satisfy the equation

$\partial_{x^{\prime}}u(x^{\prime},\tau_{x^{\prime}})\langle\tilde{\tau}_{x^{\prime}}\rangle-\partial_{\tau_{x^{\prime}}}u(x^{\prime},\tau_{x^{\prime}})\langle\Gamma_{x^{\prime}}\langle\tilde{\tau}_{x^{\prime}},\tilde{\tau}_{x^{\prime}}\rangle\rangle=0$
(22)

with boundary conditions
$u(x,\tilde{\tau}_{x}):=k_{x}\langle\tilde{\tau}_{x}\rangle$, whose
characteristics are geodesics. Equation (22), taking into account expression
(21), leads to the following equation for the deformation functions:

$(\partial^{2}_{x^{\prime}}H_{x}(x^{\prime}-x)-\partial_{x^{\prime}}H_{x}(x^{\prime}-x)\circ\Gamma_{x^{\prime}})\langle\tilde{\tau}_{x^{\prime}},\tilde{\tau}_{x^{\prime}}\rangle=0$
(23)

with boundary conditions

$H_{x}(0)=0,\quad\partial_{x^{\prime}}H_{x}(0)=h_{x}.$ (24)

The Cauchy problem (23), (24) has a unique solution.

Proposition 2. From the deformation coefficients $h_{x}$ and the connection
$\Gamma_{x}$, the map of the canonical deformation $H_{x}(\tilde{t}_{x})$, and
hence the canonical group $T^{gH}_{X}$, are uniquely determined.

We especially note that the condition of canonicity does not impose
restrictions on the mappings $h_{x}$ and $\Gamma_{x}$.

##  3 Group of Riemannian translations

Since all vectors tangent to the geodesic are parallel to each other, equation
(19) describes the finite parallel transport of the tangent vector
$\tau_{x^{\prime}}$ along the geodesic to the initial point $x$:

$\tau_{x||}=\tau^{\prime}_{x}:=\lambda_{x}(s\tau_{x})\langle\tau_{x^{\prime}}\rangle.$
(25)

Since $\lambda_{x}(s\tau_{x})\cong 1+s\tau_{x}\cdot\gamma_{x}$, this agrees
with the parallel transport rule (15) of an arbitrary vector with an
infinitesimal displacement, and takes place due to the fulfillment of the
composition law for transports of vectors tangent to geodesics along the same
geodesics:
$\lambda_{x}((s+s^{\prime})\tau_{x})\langle\tau_{x^{\prime\prime}}\rangle=\lambda_{x}(s\tau_{x})\langle\lambda_{x^{\prime}}(s^{\prime}\tau_{x^{\prime}})\langle\tau_{x^{\prime\prime}}\rangle\rangle$,
due to which the finite parallel transport of the tangent vector (25)
coincides with the integral sequence of infinitesimal transports (as
classically the finite parallel transport is defined).

Note that for vectors $\theta$ transverse to the tangents to the geodesic, a
similar relation
$\lambda_{x}((s+s^{\prime})\tau_{x})\langle\theta_{x^{\prime\prime}}\rangle=\lambda_{x}(s\tau_{x})\langle\lambda_{x^{\prime}}(s^{\prime}\tau_{x^{\prime}})\langle\theta_{x^{\prime\prime}}\rangle\rangle$
in a curved space not fulfilled. Indeed, since the vector $\theta$ are
arbitrary, this relation with help of equation (5) leads to the equality
$\partial^{\prime}\varphi_{x}(s\tau_{x},s^{\prime}\tau_{x^{\prime}})=\lambda_{x}(s\tau_{x})$,
which immediately implies $\rho_{x}=0$, and hence $R_{x}=0$. Thus, in a curved
space, the finite $\lambda$-transport of the vector $\theta$ transversal to
the tangent to the geodesic, defined by the formula
$\theta^{\prime}_{x}:=\lambda_{x}(s\tau_{x})\langle\theta_{x^{\prime}}\rangle$,
is not a parallel transport along geodesic, despite the fact that for
infinitesimal displacements (15) it it is, and this is precisely due to the
fact that in this case the compositional law of transport is violated. In this
case, the curvature of the Riemannian space acts as a measure of the deviation
of geodesics.

According to Riemann, “measurement consists in the sequential attachment of
comparable quantities, therefore the possibility of measurements is due to the
presence of some way to transport one quantity, taken as a unit of scale, to
another quantity” [1]. This is exactly what the canonical group $T^{gH}_{X}$
does. Indeed, by virtue of the canonicity, the operation of n-fold transport
of the tangent vector $\tau_{x}$ along the geodesic gives

$x_{n}=x+K_{x}(\tau_{x})+K_{x_{1}}(\tau_{x_{1}})+\ldots+K_{x_{n-1}}(\tau_{x_{n-1}})=x+K_{x}(n\tau_{x}).$

Thus, the geodesic connecting the points $x$ and
$x^{\prime}=x+K_{x}(s\tau_{x})$ contains $s$ vectors $\tau$ when they are
transported along themselves, and the affine parameter of the geodesic $s$
gives the measure of its length on the scale of the vector $\tau$.

Specifying the length of a vector $\tau_{x}\in T$, regardless of its direction
and location $\tau^{2}_{x}:=\eta\langle\tau_{x},\tau_{x}\rangle$, i.e.,
specifying the Euclidean metric $\eta$ in the space $T$, allows one to compare
the lengths of arbitrary geodesics, which implements the Riemann length
principle. Moreover, in this way the structure of a Riemannian space with
metric $g_{x}\langle\tilde{\tau}_{x},\tilde{\tau}_{x}\rangle:=\eta\langle
h_{x}\langle\tilde{\tau}_{x}\rangle,h_{x}\langle\tilde{\tau}_{x}\rangle\rangle$
is setting on $X$. The vector $\tilde{\tau}_{x}$, tangent to the geodesic,
acts as a scale unit, and therefore it is natural to require that its length
be invariable when transported along the geodesic.

Definition 2. The canonical gauge translation group, for which the condition
of preserving the length of the unit of the scale fulfilled when it is
transported along itself
$\eta\langle\tau_{x},\tau_{x}\rangle=\eta\langle\tau_{x^{\prime}},\tau_{x^{\prime}}\rangle$,
is called the group of Riemannian translations and denoted as $RT$.

Taking into account criterion (19), this condition can be represented in the
form
$G_{x}(s\tau_{x})\langle\tau_{x^{\prime}},\tau_{x^{\prime}}\rangle=\eta\langle\tau_{x^{\prime}},\tau_{x^{\prime}}\rangle$,
where
$G_{x}(t)\langle\tau,\tau\rangle:=\eta\langle\lambda_{x}(t)\langle\tau\rangle,\lambda_{x}(t)\langle\tau\rangle\rangle$,
or, using expression (20), as

$\eta\langle\partial_{x^{\prime}}H_{x}(x^{\prime}-x)\langle\tilde{\tau}_{x^{\prime}}\rangle,\partial_{x^{\prime}}H_{x}(x^{\prime}-x)\langle\tilde{\tau}_{x^{\prime}}\rangle\rangle=g_{x^{\prime}}\langle\tilde{\tau}_{x^{\prime}},\tilde{\tau}_{x^{\prime}}\rangle.$
(26)

Differentiation of equation (26) with respect to $x^{\prime}$ at
$x^{\prime}\rightarrow x$ gives the condition of compatibility the connection
with the metric

$2g_{x}\langle\tilde{\tau},\Gamma_{x}\langle\tilde{t},\tilde{\tau}\rangle\rangle=\partial
g_{x}\langle\tilde{t},\tilde{\tau},\tilde{\tau}\rangle,$ (27)

or taking into account (16)

$\eta\langle\tau,\gamma_{x}\langle t,\tau\rangle\rangle=0,$ (28)

where all vectors are taken at the point $x$. Condition (27), together with
the fact that the mapping $\Gamma_{x}$ is symmetric, allows us to express it
in terms of the derivative of the metric $\partial g_{x}$:

$g_{x}\langle\tilde{t},\Gamma_{x}\langle\tilde{\tau},\tilde{\tau}\rangle\rangle=\partial
g_{x}\langle\tilde{\tau},\langle\tilde{t},\tilde{\tau}\rangle\rangle-\frac{1}{2}\partial
g_{x}\langle\tilde{t},\langle\tilde{\tau},\tilde{\tau}\rangle\rangle,$

and condition (28) together with (10) leads to an expression for the mapping
$\gamma_{x}$ of the group $RT$ in terms of its structure operator

$\eta\langle\tau,\gamma_{x}\langle
t,\tau^{\prime}\rangle\rangle=\frac{1}{2}(\eta\langle
t,C_{x}\langle\tau,\tau^{\prime}\rangle\rangle+\eta\langle\tau,C_{x}\langle
t,\tau^{\prime}\rangle\rangle-\eta\langle\tau^{\prime},C_{x}\langle
t,\tau\rangle\rangle),$

whence it follows that in this case the connection $\Gamma_{x}$ is formed by
the Christoffel symbols, and $\gamma_{x}$ by the Ricci rotation coefficients
and, therefore, are uniquely determined by the frame fields $e_{x}$ which
defined on $X$ by the action of the group $RT$.

Condition (26) does not impose restrictions on the deformation coefficients
$h_{x}$ (and hence the fields $e_{x}$); therefore, taking into account Theorem
1 and Proposition 2, we arrive at the following statement.

Theorem 2. The group of Riemannian translation $RT$, by its action on the
manifold $X$, defines the structure of a Riemannian space and an orthonormal
frame field $e_{x}$, by which it is uniquely determined. An arbitrary
Riemannian structure on a manifold $X$ can be defined in this way.

The requirement that the length of the vector $\tau$ tangent to the geodesic
be preserved under finite $\lambda$-transport does not provide a similar
property for an arbitrary vector $\theta$.

Proposition 3. The requirement that the length of an arbitrary vector be
preserved under a finite $\lambda$-transport $G_{x}(s\tau)=\eta$ is a
condition of flat space.

The statement follows from the relation

$\frac{d^{2}}{ds^{2}}G_{x}(s\tau)\langle\theta,\theta\rangle|_{s=0}=\frac{4}{3}\eta\langle\theta,R_{x}\langle\tau,\tau,\theta\rangle\rangle,$

which holds for the group RT [12].

Thus, in spite of the fact that the group $RT$ contains complete information
on the geometrical structure of the Riemannian space and defines it by its
action on $X$, it leaves no invariants either in $X$ or in $TX$, which does
not allow us to consider group $RT$ as the principal group of Riemannian
space. The group of Riemannian translation $RT$ is the source of geometric
objects in Riemannian space: its generators form an orthonormal frame field,
which defines a metric on the manifold $X$; parameters are vector fields on
$X$. The group $RT$ allows one to measure the lengths of arbitrarily located
curves by moving a unit scale along them and thus implements the principle of
length (B. Riemann).

##  4 Group of motions of tangent bundle of Riemannian space

The action of group of Riemannian translations $RT$ defines on the manifold
$X$ not only a Riemannian structure, that is, a metric $g_{x}$, but also a
fixed field of orthonormal frames $e_{x}=k_{x}\cdot\partial_{x}$. Therefore,
the same Riemannian structure on $X$ is given by all groups $RT$ whose frame
fields are related by transformations from the gauge rotation group
$\tilde{r}_{x}\in R^{g}$:

$e^{\prime}_{x^{\prime}}=\tilde{r}^{-1}_{x}\cdot
e_{x^{\prime}},\quad\eta\langle
r_{x}\langle\tau\rangle,r_{x}\langle\tau\rangle\rangle=\eta\langle\tau,\tau\rangle\
\forall x\in X,\ \tau\in T,$ (29)

at which the deformation coefficients of the group $RT$ transform according to
the law $h^{\prime}_{x^{\prime}}=\tilde{r}_{x}\circ h_{x^{\prime}}$, which
leaves the metric $g_{x}$ unchanged. Transformations (29) are taken non-local
($x^{\prime}\neq x$!), which determines a moving frame (according to E.
Cartan): here $e^{\prime}_{x^{\prime}}$ is interpreted as a frame transported
from point $x$ to point $x^{\prime}$ and arbitrarily turned by the value
$\tilde{r}_{x}^{-1}$, given at the point $x$, where the translation value
$\tilde{t}_{x}=x^{\prime}-x$ given, too.

The set of such transformations forms a generalized gauge group
$G^{g}_{X}=T^{g}_{X}\times)R^{g}$ [8], which has the structure of a semidirect
product $\times)$ of its subgroups, with parameters
$\tilde{\vartheta}_{x}=(\tilde{t}_{x},\tilde{r}_{x})$ and the multiplication
law
$(\tilde{\vartheta}\times\tilde{\vartheta}^{\prime})_{x}=:\tilde{\Phi}_{x}(\tilde{\vartheta}_{x},\tilde{\vartheta}^{\prime}_{x^{\prime}})$:

$(\tilde{\vartheta}\times\tilde{\vartheta}^{\prime})^{T}_{x}=\tilde{\Phi}_{x}^{T}(\tilde{\vartheta}_{x},\tilde{\vartheta}^{\prime}_{x^{\prime}}):=\tilde{t}_{x}+\tilde{t}^{\prime}_{x^{\prime}},$
(30)
$(\tilde{\vartheta}\times\tilde{\vartheta}^{\prime})^{R}_{x}=\tilde{\Phi}_{x}^{R}(\tilde{\vartheta}_{x},\tilde{\vartheta}^{\prime}_{x^{\prime}}):=\tilde{r}_{x}\circ\tilde{r}^{\prime}_{x^{\prime}},$
(31)

where $x^{\prime}=x+\tilde{t}_{x}$ defines the action of the group $G^{g}_{X}$
on $X$, and the indices $T$ and $R$ indicate the canonical projection of the
elements of the group $G^{g}_{X}$ onto its factors $T^{g}_{X}$ and $R^{g}$.

The condition $\tau^{\prime}_{x}\cdot
e^{\prime}_{x^{\prime}}=\tau_{x^{\prime}}\cdot e_{x^{\prime}}$, which follows
from the interpretation of transformations (29) as frame motions, determines
the action of the group $G^{g}_{X}$ on the parameters of the group $RT$:

$\tau^{\prime}_{x}=\tilde{r}_{x}\langle\tau_{x^{\prime}}\rangle,$ (32)

that is, in the tangent bundle $TX$. Thus, the group $G^{g}_{X}$ describes
nonlocal automorphisms of the group $RT$ that leave the Riemannian structure
it defines on $X$ unchanged.

Transformations (29) and (32) in the absence of rotations $\tilde{r}_{x}=1$
describe the componentwise identification of vectors given at different
points, which corresponds to the case of flat space and does not agree with
the Riemannian structure defined on $X$ by the action of the group $RT$. To
implement such a match, we deform the group $G^{g}_{X}$.

Let the deformed group $G^{g\bar{H}}_{X}=T^{g\bar{H}}_{X}\times)R^{g}$ be
parameterized by the pairs $\vartheta_{x}=(t_{x},r_{x})$ in such a way that

$\tilde{t}_{x}=\bar{K}_{x}(t_{x}),$ (33)
$\tilde{r}_{x}=r_{x}\circ\pi_{x}(t_{x}),$ (34)

where $\bar{K}_{x}=\bar{H}^{-1}_{x}$ and $\pi_{x}(t)\in R^{g}$, that is,

$\eta\langle\pi_{x}(t)\langle\tau\rangle,\pi_{x}(t)\langle\tau\rangle\rangle=\eta\langle\tau,\tau\rangle,\quad\pi_{x}(0)=1\
\forall x\in X,\ t,\tau\in T.$ (35)

Formula (33) describes an independent deformation of the subgroup
$T^{g}_{X}\subset G^{g}_{X}$ to $T^{g\bar{H}}_{X}$ using the deformation map
$\bar{H}_{x}$, and (34), being substituted in (32), ensures the rotation
$\pi_{x}(t_{x})$ vectors under translations:

$\tau^{\prime}_{x}=r_{x}\langle\pi_{x}(t_{x})\langle\tau_{x^{\prime}}\rangle\rangle,$
(36)

where $x^{\prime}=x+\bar{K}_{x}(t_{x})$ defines the action of the group
$G^{g\bar{H}}_{X}$ on $X$, and (36) on $TX$.

The multiplication law
$(\vartheta\times\vartheta^{\prime})_{x}=:\Phi_{x}(\vartheta_{x},\vartheta^{\prime}_{x^{\prime}})$
of the group $G^{g\bar{H}}_{X}$ is determined by its isomorphism (33), (34) to
the group $G^{g}_{X}$ and the multiplication law (30), (31) of the group
$G^{g}_{X}$:

$(\vartheta\times\vartheta^{\prime})^{T}_{x}=\Phi_{x}^{T}(\vartheta_{x},\vartheta^{\prime}_{x^{\prime}}):=\bar{\varphi}_{x}(t_{x},t^{\prime}_{x^{\prime}})=\bar{H}_{x}(\bar{K}_{x}(t_{x})+\bar{K}_{x^{\prime}}(t^{\prime}_{x^{\prime}})),$
$(\vartheta\times\vartheta^{\prime})^{R}_{x}=\Phi_{x}^{R}(\vartheta_{x},\vartheta^{\prime}_{x^{\prime}}):=r_{x}\circ\pi_{x}(t_{x})\circ
r^{\prime}_{x^{\prime}}\circ\pi_{x^{\prime}}(t^{\prime}_{x^{\prime}})\circ\pi^{-1}_{x}(\bar{\varphi}_{x}(t_{x},t^{\prime}_{x^{\prime}})).$
(37)

Transformation (36) in the case of pure displacement, i.e., with
$\vartheta_{x}=(t_{x},1)$, is called the $\pi$-transport. In the infinitesimal
case, it takes the form
$\tau^{\prime}_{x}=\tau_{x}+t_{x}\cdot\bar{\nabla}_{x}\tau_{x}$, where
$\bar{\nabla}_{x}:=\bar{e}_{x}+\bar{\gamma}_{x}$,
$\bar{e}_{x}:=\bar{k}_{x}\cdot\partial_{x}$,
$\bar{k}_{x}:=\partial\bar{K}_{x}(0)$, $\bar{\gamma}_{x}:=\partial\pi_{x}(0)$,
and corresponds to the parallel transport of the vector $\tau_{x}$ from the
point $x^{\prime}=x+\bar{k}_{x}\langle t_{x}\rangle$ to the point $x$ in the
affinely connected space with the connection $\bar{\gamma}_{x}$ given in the
frame $\bar{e}_{x}$. In this sense, this geometric structure is defined on $X$
by the action (36) of the group $G^{g\bar{H}}_{X}$ in $TX$.

We require that the consistency condition $\bar{\nabla}_{x}=\nabla_{x}$ (or
$\bar{e}_{x}=e_{x}$, $\bar{\gamma}_{x}=\gamma_{x}$) be satisfied, which
ensures coincidence at the infinitesimal level of $\pi$-transports with
$\lambda$-transports, and hence the coincidence of the geometric structure
given by the group $G^{g\bar{H}}_{X}$, to the structure given by the original
group $RT$, whose nonlocal automorphisms are described by the group
$G^{g\bar{H}}_{X}$. Note that, due to requirement (35), condition of metric
compatibility (28) is satisfied automatically, and the condition
$\bar{\gamma}_{x}=\gamma_{x}$, due to the accepted equality
$\bar{e}_{x}=e_{x}$, is reduced to the nontorsionity condition

$C_{x}\langle l,l^{\prime}\rangle:=\bar{\gamma}_{x}\langle
l,l^{\prime}\rangle-(l\leftrightarrow l^{\prime})$

Definition 3. The deformed group $G^{g\bar{H}}_{X}$ of nonlocal automorphisms
of the group of Riemannian translations $RT$ preserving the Riemannian
structure defined on $X$ by the action of the group $RT$, for which the
consistency condition $\bar{\nabla}_{x}=\nabla_{x}$ is satisfied, will be
called the group of motions of tangent bundle of Riemannian space and denoted
as $DR$.

In terms of the mappings $L_{x}(\vartheta):=r\circ\pi_{x}(t)$, the
compositional law of transformations (36)
$\tau^{\prime}_{x}=L_{x}(\vartheta)\langle\tau_{x^{\prime}}\rangle$, written
for constant parameters $\vartheta=(t,r)$, takes the form

$L_{x}(\Phi_{x}(\vartheta,\vartheta^{\prime}))=L_{x}(\vartheta)\circ
L_{x^{\prime}}(\vartheta^{\prime}),$ (38)

where $x^{\prime}=x+\bar{K}_{x}(t)$. Introducing the notation

$\Lambda_{x}(\vartheta):=\partial_{\vartheta^{\prime}}\Phi_{x}(\vartheta,\vartheta^{\prime})|_{\vartheta^{\prime}=(0,1)},\quad
E_{x}:=\partial_{\vartheta}L_{x}(\vartheta)|_{\vartheta=(0,1)},$

where $\partial_{\vartheta}:=\partial/\partial\vartheta$, and differentiating
equation (38) with respect to $\vartheta^{\prime}$ at
$\vartheta^{\prime}=(0,1)$, we obtain an analogue of the Lie equation for the
group $DR$ of transformations (36):

$\Lambda_{x}(\vartheta)\cdot\partial_{\vartheta}L_{x}(\vartheta)=L_{x}(\vartheta)\circ
E_{x^{\prime}}.$ (39)

The integrability condition for equation (39) is the equation

$[D_{x},D_{x}]=\Sigma_{x}\cdot D_{x},$ (40)

written by us in terms of the commutator of the generators $D_{x}$ of the
group $DR$ of transformations (36), which are determined by the relation

$D_{x}\tau_{x}:=\partial_{\vartheta}(L_{x}(\vartheta_{x})\langle\tau_{x^{\prime}}\rangle)|_{\vartheta=(0,1)},$

and its structure operator

$\Sigma_{x}\langle\xi,\xi^{\prime}\rangle:=\partial_{\vartheta}\Lambda_{x}(\vartheta)\langle\xi,\xi^{\prime}\rangle|_{\vartheta=(0,1)}-(\xi\leftrightarrow\xi^{\prime}),$
(41)

where in this case $\xi$ and $\xi^{\prime}$ are the vectors of the space
tangent to the unit of the group $G=T\otimes R$. Equation (40) generalizes the
Maurer – Cartan equation of the theory of finite-dimensional Lie groups of
transformations for the group $DR$ of transformations (36).

The generators $D_{x}$ split into translation generators, which coincide with
the covariant derivatives $\nabla_{x}$ of vector fields, and also
$x$-independent generators $A$ of rotations of vectors in the Euclidean space
$T$.

A special case of (40) is the equation

$[\nabla_{x},\nabla_{x}]=\Sigma_{x}{}^{T}_{TT}\cdot\nabla_{x}+\Sigma_{x}{}^{R}_{TT}\cdot
A,$ (42)

for the commutator of translation generators, where the subscript $T$ means
the restriction of the mapping to the subspace $T$. Direct calculations using
the defining formula (41) give the following expressions for the components of
the structure operator of the group $DR$: $\Sigma_{x}{}^{T}_{TT}=C_{x}$,
$\Sigma_{x}{}^{R}_{TT}=R_{x}$, as a result of which equation (42) takes the
form of the structural equation of a Riemannian space

$[\nabla_{x},\nabla_{x}]=C_{x}\cdot\nabla_{x}+R_{x}$ (43)

with the curvature tensor $R_{x}$ in the frame $e_{x}$, whose anholonomic
object is $C_{x}$ (here it is taken into account that $R_{x}=R_{x}\cdot A$).
Equation (43) using equation (9) $[e_{x},e_{x}]=C_{x}\cdot e_{x}$ is reduced
to (12).

Jacobi identity for the group $DR$

$D_{x}\Sigma_{x}\langle\xi,\xi^{\prime},\xi^{\prime\prime}\rangle+\Sigma_{x}\langle\xi,\Sigma_{x}\langle\xi^{\prime},\xi^{\prime\prime}\rangle\rangle+cycl(\xi,\xi^{\prime},\xi^{\prime\prime})=0$

is the integrability condition for the Maurer – Cartan equation (40), and when
it is restricted to the translation $\xi\rightarrow l$ reduces to the Bianchi
identity.

Thus, it is proved

Theorem 3. The translation generators of the action of the group of motions of
the tangent bundle of the Riemannian space $DR$ in its tangent bundle $TX$ are
covariant derivatives of vector fields $\nabla_{x}=e_{x}+\gamma_{x}$, the
structure operator of the group $DR$ has as components the object of
anholonomity $C_{x}$ of the orthonormal frame field $e_{x}$ and the curvature
tensor $R_{x}$ in it, and the structure equation of the Riemannian space
$[\nabla_{x},\nabla_{x}]=C_{x}\cdot\nabla_{x}+R_{x}$ is a necessary condition
for the existence of the group $DR$ which define by its action in the bundle
$TX$ this structure on $X$.

We see that both $DR$ and $RT$ define the structure of a Riemannian space on
$X$, but in two different ways. If for group $DR$ an infinitesimal action in
the tangent bundle $TX$ is sufficient and structure equation (43) appears as a
component of the Maurer – Cartan equation which ensure the existence of the
group $DR$, then for group $RT$ it is required to specify its action in $X$
already with an accuracy of at least the second order in displacements, which
generates both the bundle $TX$ itself and the action of the group $RT$ in it,
and the structure equation of the Riemannian space (12) appears here already
after the differentiation of the Maurer – Cartan equation for the group $RT$.
The reason for this is that the groups $DR$ and $RT$ describe different
aspects of the curvature of the Riemannian space, namely, in the case of group
$DR$, curvature acts as a measure of the noncommutativity of its generators
(or the measure of the rotation of the vector when it is swept along a closed
contour), and in the case of group $RT$, curvature acts as a measure of
geodesic deviation.

We emphasize that in the proposed group-theoretic approach, the curvature is
not added to the structural equation of a flat space for reasons of
anholonomicity of curved space, as do E. Cartan, but is a characteristic of a
group that defines a given geometric structure with variable curvature. In the
case of the group $DR$, this characteristic is its structure operator, which
is determined by the antisymmetric part of the coefficients in the second
order of the expansion of the multiplication law of the group $DR$ with
respect to parameters, and in the case of group $RT$, it is the curvature
operator, which is determined by the antisymmetric part of the coefficients
already in the third order of the expansion of the multiplication law of the
group $RT$. Thus, the anholonomicity of the space appears upon deformation of
the generalized gauge group of the corresponding flat space, as an isomorphic
transition to the deformed group, which preserves the group structure of the
transformations used, including their associativity.

In the orthonormal frame bundle (in principal bundle $RX$), the group of
motions of the tangent bundle of the Riemannian space $DR$ acts by the formula

$e^{\prime}_{x^{\prime}}=\pi^{-1}_{x}(t_{x})\cdot r^{-1}_{x}\cdot
e_{x^{\prime}},$

This formula follows from (29) (taking into account deformation (34)),
determines the movable frame of E. Cartan [10] and allows one to define
connection by means of a movable frame (with r = 1):

$\nabla_{e_{x}}e_{x}:=\lim_{t\rightarrow
0}\frac{e_{x}-e^{\prime}_{x}}{t}=\gamma_{x}\cdot e_{x}.$

Since $r_{x},\pi_{x}\in R^{g}$, transformations from the group of motions of
the tangent bundle of the Riemannian space $DR$ preserve the lengths of
vectors and the angles between them, and therefore allow one to establish the
equality of geometric figures by superimposing them, even if these figures are
located at different points of the space $X$. Here, under geometric figures we
mean configurations of vectors defined in tangent spaces. Thus, $DR$ realizes
the principle of equality and is the principal group of Riemannian space
(according to F. Klein).

##  5 Group of parallel transports of Riemannian space

Since the infinitesimal action of the group $DR$ is sufficient to define the
geometric structure, the same Riemannian structure is given by the set of
groups $DR$ differing in higher orders of action in the bundles $TX$ or $RX$.
On the other hand, the question arises - what does the finite $\pi$-transport
mean? Along what curve is it carried out? These questions are similar to the
questions about the meaning of finite $\lambda$-transports in the case of the
deformed group of diffeomorphisms $T^{gH}_{M}$, which were successfully
clarified for it using the concept of canonicity.

For the group $DR$, which is broader than the group $T^{gH}_{M}$, the concept
of canonicity is somewhat broadened, but equally fruitful.

Definition 4. The group of motions of the tangent bundle of the Riemannian
space $DR$ will be called the group of parallel transports of Riemannian space
and denoted as $DP$ if it satisfies the extended canonicity condition, namely,
for any two points $x,x^{\prime}\in X$, there exists a smooth parametric curve
$x^{\prime}=x(s)$, $x=x(0)$, such that

$((s\tau,1)\times(s^{\prime}\tau,1))_{x}=(\bar{\varphi}_{x}(s\tau_{x},s^{\prime}\tau_{x^{\prime}}),1)=((s+s^{\prime})\tau_{x},1),$

where
$\tau_{x^{\prime}}=h_{x^{\prime}}\langle\tilde{\tau}_{x^{\prime}}\rangle$,
$\tilde{\tau}_{x^{\prime}}=\dot{x}^{\prime}$.

This definition, firstly, requires the subgroup $T^{g\bar{H}}_{M}\subset DP$
to be canonic, and, due to the consistency condition $\bar{e}_{x}=e_{x}$ and
Theorem 2, the coincidence of the deformation mappings $\bar{H}_{x}=H_{x}$,
and hence the coincidence of the entire group $T^{g\bar{H}}_{M}$ with the
original group of Riemannian translation $RT$, which also implies the
existence of the desired curves and their coincidence with the geodesics given
by the group $RT$. Second, as follows from formula (37), it leads to the
fulfillment of the composition law for $\pi$-transports of arbitrary vectors
along geodesics:
$\pi_{x}((s+s^{\prime})\tau_{x})=\pi_{x}(s\tau_{x})\circ\pi_{x^{\prime}}(s^{\prime}\tau_{x^{\prime}})$.
Since for infinitesimal displacements, the $\pi$-transport is a parallel
transport of vectors, the fulfillment of this law leads to the fact that the
finite $\pi$-transport of arbitrary vectors coincides with the integral
sequence of their infinitesimal parallel transports along the geodesic
connecting points $x$ and $x^{\prime}$ (which determined the name of the group
$DP$ ).

Due to the consistency condition, in the group of parallel transports $DP$, as
in all groups $DR$, the infinitesimal $\pi$-transports and
$\lambda$-transports of arbitrary vectors coincide, while the finite
transports in a curved space are different. But only in the group $DP$ for
vectors tangent to geodesics, both finite $\pi$-transports and
$\lambda$-transports lead to the same results.

To find the equation defining the mapping $\pi_{x}$, note that the functions

$u(x^{\prime},\theta_{x^{\prime}}):=\pi_{x}(H_{x}(x^{\prime}-x))\langle\theta_{x^{\prime}}\rangle,$
(44)

due to the fact that along geodesics connecting the pairs of points $x$ and
$x^{\prime}$ $u(x^{\prime},\theta_{x^{\prime}})=\theta_{x}=const$, are the
first integrals of the system of equations
$\dot{x}^{\prime}=k_{x}\langle\tau_{x^{\prime}}\rangle$,
$\dot{\theta}_{x^{\prime}}=-\gamma_{x^{\prime}}\langle\tau_{x^{\prime}},\theta_{x^{\prime}}\rangle$,
and therefore satisfy the partial differential equation:

$k_{x}\cdot\partial_{x^{\prime}}u(x^{\prime},\theta_{x^{\prime}})\langle\tau_{x^{\prime}}\rangle-\partial_{\theta_{x^{\prime}}}u(x^{\prime},\theta_{x^{\prime}})\langle\gamma_{x^{\prime}}\langle\tau_{x^{\prime}},\theta_{x^{\prime}}\rangle\rangle=0,$

substitution into which expression (44) leads to the equation

$k_{x^{\prime}}\cdot\partial_{x^{\prime}}\pi_{x}(H_{x}(x-x^{\prime}))-\pi_{x}(H_{x}(x-x^{\prime}))\circ\gamma_{x^{\prime}})\langle\tau_{x^{\prime}},\theta_{x^{\prime}}\rangle=0,$

giving, under the condition $\pi_{x}(0)=1$ and arbitrary initial values
$\tau_{x},\theta_{x}\in T$, a unique solution for the map $\pi_{x}$. Thus, the
deformation (33), (34) with the help of which the group of parallel transports
$DP$ is constructed, is uniquely determined.

Proposition 4. For a given orthonormal frame field $e_{x}$, the group of
parallel transports of Riemannian space $DP$ is uniquely determined.

Thus, we can say that the group of parallel transports $DP$ is univocal
extension of the group of Riemannian translations $RT$ to the possibility of
taking into account local rotations from $R^{g}$. Uniqueness is achieved,
first of all, due to the canonicity condition, which, from a geometric point
of view, is reduced to fixing geodesics as curves along which finite
transports are carried out, as well as imposing a consistency condition. All
groups $DP$ correspond to the same Riemannian structure, if for them frames
$e_{x}$ are related by transformations from $R^{g}$.

Theorem 4. The group of parallel transports of Riemannian space $DP$, acting
on the manifold $X$ as the group of Riemannian translations $RT$ (with the
inefficiency kernel $R^{g}$), defines the structure of a Riemannian space on
$X$, making it possible to measure the lengths of arbitrary curves by moving a
unit scale, which implements the principle of length.

Acting in the tangent bundle $TX$, group $DP$ preserves the lengths of vectors
and the angles between them, making it possible to establish the congruence of
geometric shapes by superimposing them, which implements the principle of
equality.

So the group of parallel transports of Riemannian space $DP$ combines both
principles of the foundations of geometry, thus overcoming the Riemann-Klein
antagonism.

## Conclusions

The main results of this article are as follows:

the group of Riemannian translations $RT$ is the source of geometric objects
of Riemannian space and implements the length principle;

the group of motions of tangent bundle of Riemannian space $DR$, as the group
of nonlocal automorphisms of the group $RT$ compatible with $RT$, is the
principal group of Riemannian space, which realizes F. Klein’s Erlangen
program for it - the equality principle;

the group of parallel transports $DP$, as a canonical group $DR$, containing
the group of Riemannian translations $RT$ as a subgroup, combines both
principles and coordinates them.

## References

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