Reduction of Bundles, Connection, Curvature, and Torsion of the Centered Planes Space at Normalization

Abstract

The space $\Pi$ of centered m-planes is considered in projective space $P_n$. A principal bundle is associated with the space $\Pi$ and a group connection is given on the principal bundle. The connection is not uniquely induced at the normalization of the space $\Pi$. Semi-normalized spaces ${\Pi }^1$, ${\Pi }^2$ and normalized space ${\Pi }^{1,2}$ are investigated. By virtue of the Cartan–Laptev method, the dynamics of changes of corresponding bundles, group connection objects, curvature and torsion of the connections are discovered at a transition from the space $\Pi$ to the normalized space ${\Pi }^{1,2}$.

1. Introduction

This paper refers to the field of differential geometry, or, more precisely, to the theory of differentiable manifolds equipped with various “geometric structures” [1], such as connection, curvature and torsion. We use the following methods: the moving frame method (the E. Cartan method [2] of differential-geometric research) and the G. F. Laptev method of extensions and envelopments, which includes the Cartan moving frame method and gives a universal character to the first one. Universality and efficiency of the Cartan method were shown in many papers. Upon use of this method, a research of geometry of a manifold with geometric structures fixed on it is reduced to study of geometry of other manifolds (total space of frames above the given manifold or subbundles of the bundle). Thus, automatically there is an analytical apparatus that is most adapted to research of the initial structure [3]. The method of extensions and envelopments is based on the invariant differential-algebraic apparatus of structure differential forms of considered bundles [4]. In this paper, the Cartan–Laptev method is applied to research of the centered m-planes space in projective space $P_n$.

The connection theory (see, e.g., [5,6]) has an important place in differential geometry. A lot of research devoted to the geometry of planes manifolds in classical spaces includes studying connections. The connection theory also plays a fundamental role in physics.

Normalization of a manifold [7] of centered planes in projective space can be defined by an analogy with the Norden normalization [8]. A. P. Norden has described the normalization of a surface. A surface $X_m$ can be considered as an m-dimensional manifold of m-dimensional centered planes $T_m$ [9]. Yu. G. Lumiste [10] has entered a similar normalization of a manifold of m-planes in projective space. An analogue of this normalization is used in this paper. Thus, in our case, the normal of the first kind (the first normal) is a subspace $N_{n-m}$ of $P_n$ having only one common point with a centered m-plane $T_m$; and the normal of the second kind (the second normal) is a subspace $N_{m-1}$ of the centered m-plane $T_m$ not passing through its centre [11]. Moreover, we will also use a reduction that is frequently applied in geometry (see, e.g., [4,12,13,14,15,16,17]).

This paper is a continuation of author’s research [18,19]. Our purpose is to give the full analysis of the dynamics of changes of bundles, connection, curvature and torsion at transition from the space of centered planes to the normalized space.

The timeliness of the present paper is caused by the facts that the space of centered planes is a set of all m-dimensional centered planes, which we may say about a communication with the Grassmann manifold (the set of all m-planes) [20]. It is important to emphasize that the Grassmann manifold plays a key role in topology and geometry as the base space of an universal vector bundle. Moreover, $Gr(1,n+1)$ is projective space $P_n$.

2. Analytical Apparatus

Projective space $P_n$ can be presented as the quotient space $L_{n+1}/\sim$ of a vector space $L_{n+1}$ by equivalence relation (collinearity) ∼ of nonzero vectors, i.e., $P_n=(L_{n+1}\backslash \left\{0\right\})/\sim$ (see, e.g., [21]). Thus, we can set the quotient map by

$$L_{n+1}\backslash \left\{0\right\}\to P_n.$$

As is known [11], a projective frame in the space $P_n$ is a system consisting of points $A_{I^{\prime }}$, $I^{\prime }=0,\dots ,n$, and the unity point E. In the vector space $L_{n+1}$, linearly independent vectors $e_{I^{\prime }}$ correspond to the points $A_{I^{\prime }}$ and the vector $e={\sum }_{I^{\prime }=0}^n{e}_{I^{\prime }}$ corresponds to the point E. These vectors are defined up to a common factor in $L_{n+1}$. The unity point E is given along with the basis points $A_{I^{\prime }}$, though we might not mention it each time.

It is supposed that a frame in the vector space $L_{n+1}$ is normalized, i.e., $e_0\wedge e_1\wedge \dots \wedge e_n=1$, where ∧ sets an exterior product.

The equations of infinitesimal displacements of the moving frame in $P_n$ can be written in the following way:

$$d{A}_{I^{\prime }}={\theta }_{I^{\prime }}^{J^{\prime }}{A}_{J^{\prime }},$$

$I^{\prime }=0,\dots ,n$, with the condition of normalization $A_0\wedge A_1\wedge \dots \wedge A_n=1$. Here, d denotes ordinary differentiation in $P_n$. The forms ${\theta }_{I^{\prime }}^{J^{\prime }}$ are linear differential forms; they depend on parameters u (defining a location of the frame) and their differentials $du$.

The forms ${\theta }_{I^{\prime }}^{J^{\prime }}$ are connected by the relation ${\theta }_0^0+{\theta }_1^1+\dots +{\theta }_n^n=0$. This condition is also necessary for the number of linearly independent forms ${\theta }_{I^{\prime }}^{J^{\prime }}$ that became equal to the number of parameters on which the group of projective transformations of space $P_n$ depends.

The structure equations of projective space $P_n$ have the form

$$D{\theta }_{I^{\prime }}^{J^{\prime }}={\theta }_{I^{\prime }}^{K^{\prime }}\wedge {\theta }_{k^{\prime }}^{J^{\prime }},$$

where D is a symbol of exterior derivative.

By the condition ${\theta }_{I^{\prime }}^{I^{\prime }}=0$ from the linear group $GL(n+1)$, it is possible to determine the special linear group $SGL(n+1)$ [21] acting effectively in $P_n$.

Introducing the following new forms (see, e.g., [9,10]) ${\omega }_{J^{\prime }}^{I^{\prime }}={\theta }_{J^{\prime }}^{I^{\prime }}-{\delta }_{J^{\prime }}^{I^{\prime }}{\theta }_0^0$ and fixing the index $I^{\prime }=\{0,I\}$, $I=1,\dots ,n$, we can expand the forms ${\omega }_{J^{\prime }}^{I^{\prime }}$ as

$${\omega }_0^I={\theta }_0^I,{\omega }_J^I={\theta }_J^I-{\delta }_J^I{\theta }_0^0,{\omega }_I^0={\theta }_I^0({\omega }_0^0=0).$$

The formulas of infinitesimal displacements can be written in more detail:

$$dA=\theta A+{\omega }^I{A}_I,d{A}_I=\theta A_I+{\omega }_I^J{A}_J+{\omega }_{I}A,$$

(1)

where $A=A_0$, ${\omega }^I={\omega }_0^I$, ${\omega }_I={\omega }_I^0$, and the form $\theta ={\theta }_0^0$ plays the role of the proportionality coefficient.

Introducing the basis forms $\omega$ and omitting, for simplicity, the index 0 in the notation of the forms ${\omega }_0^I$ and ${\omega }_I^0$, the Cartan equations can be defined

$$D{\omega }^I={\omega }^J\wedge {\omega }_J^I,D{\omega }_J^I={\omega }_J^K\wedge {\omega }_K^I+{\delta }_J^I{\omega }_K\wedge {\omega }^K+{\omega }_J\wedge {\omega }^I,D{\omega }_I={\omega }_I^J\wedge {\omega }_J,$$

(2)

where ${\omega }^I$, ${\omega }_J^I$, ${\omega }_I$ are the basis forms of the projective group $GP\left(n\right)$ acting effectively on $P_n$.

Remark 1.

We employ the inhomogeneous analytic apparatus with the derivation formulas (1) and the structure equations (2). By contrast to the homogeneous case, this apparatus is more convenient for investigation of centered planes; and it was used in the previous author’s papers [22,23,24,25].

3. The Space of Centered Planes

For the purpose of this paper, the term “space of centered planes”, denoted by $\Pi$, will be taken to mean a space of all m-dimensional centered planes $P_m^0$ in projective space $P_n$. The space $\Pi$ is a differentiable manifold and its points are m-dimensional centered planes.

Putting the vertice A of the moving frame on a m-plane $P_m$ and fixing it as a centre, we get a centered plane $P_m^0$. Putting the vertices $A_a$ of the frame on the plane $P_m^0$, we fix index ranges $1\le a,b,\dots \le m$ and $m+1\le \alpha ,\beta ,\dots \le n$. From the derivation formulas (1), we immediately get stationarity equations for the centered plane. These equations have the form ${\omega }^{\alpha }=0$, ${\omega }_a^{\alpha }=0$, ${\omega }^a=0$. The forms ${\omega }^{\alpha }$, ${\omega }_a^{\alpha }$, ${\omega }^a$ are the basis forms of the space $\Pi$; the rest forms ${\omega }_b^a$, ${\omega }_a$, ${\omega }_{\beta }^{\alpha }$, ${\omega }_{\alpha }^a$, ${\omega }_{\alpha }$ are secondary.

Remark 2.

The dimension of the space Π of centered planes differs from the dimension of the Grassmann manifold $Gr(m,n)$ [26] by the size m [22], i.e., $dim\Pi =dimGr(m,n)+m=n+m(n-m)$.

4. Principal Bundle of the Space $\mathbf{\Pi }$

The specification of the moving frame to the space $\Pi$ yields the principal bundle $G(\Pi )$, its typical fiber is a stationary subgroup G of the centered plane $P_m^0$ and base space is the space $\Pi$; in addition, thereto, dim $G=n^2-mn+m^2+n$. Total space of the bundle $G(\Pi )$ [3] is the projective group $GP\left(n\right)$ and the projection $\pi :GP\left(n\right)\to \Pi$ associates with each element of the group $GP\left(n\right)$ the plane $P_m^0$ in $\Pi$, which is invariant under the action of this element.

The basis forms ${\omega }^{\alpha }$, ${\omega }^a$, ${\omega }_a^{\alpha }$ satisfy the Cartan structure equations

$$\begin{array}{c}D{\omega }^{\alpha }={\omega }^a\wedge {\omega }_a^{\alpha }-{\omega }_{\beta }^{\alpha }\wedge {\omega }^{\beta },D{\omega }^a={\omega }^b\wedge {\omega }_b^a-{\omega }_{\alpha }^a\wedge {\omega }^{\alpha },\\ D{\omega }_a^{\alpha }=({\delta }_{\beta }^{\alpha }{\omega }_a^b-{\delta }_a^b{\omega }_{\beta }^{\alpha })\wedge {\omega }_b^{\beta }+{\omega }_a\wedge {\omega }^{\alpha }.\end{array}$$

(3)

The exterior differentials of the secondary forms are as follows:

$$\begin{array}{c}\hfill D{\omega }_b^a={\omega }_b^c\wedge {\omega }_c^a+({\delta }_c^a{\omega }_b+{\delta }_b^a{\omega }_c)\wedge {\omega }^c+{\delta }_b^a{\omega }_{\alpha }\wedge {\omega }^{\alpha }-{\omega }_{\alpha }^a\wedge {\omega }_b^{\alpha };\end{array}$$

(4)

$$\begin{array}{c}\hfill D{\omega }_a={\omega }_a^b\wedge {\omega }_b-{\omega }_{\alpha }\wedge {\omega }_a^{\alpha };\end{array}$$

(5)

$$\begin{array}{c}\hfill D{\omega }_{\beta }^{\alpha }={\omega }_{\beta }^{\gamma }\wedge {\omega }_{\gamma }^{\alpha }+{\delta }_{\beta }^{\alpha }{\omega }_a\wedge {\omega }^a+({\delta }_{\gamma }^{\alpha }{\omega }_{\beta }+{\delta }_{\beta }^{\alpha }{\omega }_{\gamma })\wedge {\omega }^{\gamma }+{\omega }_{\beta }^a\wedge {\omega }_a^{\alpha };\end{array}$$

(6)

$$\begin{array}{c}\hfill D{\omega }_{\alpha }^a={\omega }_{\alpha }^b\wedge {\omega }_b^a+{\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }^a+{\omega }_{\alpha }\wedge {\omega }^a;\end{array}$$

(7)

$$D{\omega }_{\alpha }={\omega }_{\alpha }^a\wedge {\omega }_a+{\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }.$$

Remark 3.

The principal bundle $G(\Pi )$ of the space Π contains the following five quotient bundles [22]:

1. 

$L_{m^2}\left(P_n\right)$ is the quotient bundle of linear plane frames belonging to the planes $P_m^0$, its typical fiber is the linear quotient group $L_{m^2}=GL\left(m\right)$ acting on the pencil of lines on the plane $P_m^0$ with the structure Equations (3) and (4);

2. 

$L_{{(n-m)}^2}\left(P_n\right)$ is the quotient bundle of normal linear frames; it is dual to the quotient bundle of linear plane frames; the typical fiber is the linear quotient group $L_{{(n-m)}^2}=GL(n-m)$ acting on the quotient space $P_{n-m-1}=P_n/P_m^0$ with the structure Equations (3) and (6);

3. 

$C_{m(m+1)}\left(P_n\right)$ is the quotient bundle of plane co-affine frames belonging to the plane $P_m^0$; its typical fiber is the co-affine quotient group $C_{m(m+1)}=G{A}^{*}\left(m\right)$ acting on the plane $P_m^0$ and $GL\left(m\right)\subset G{A}^{*}\left(m\right)\subset G$. This quotient bundle has the structure Equations (3)–(5);

4. 

$H_k\left(P_n\right)$ is the affine quotient bundle whose typical fiber $H_k$ ($k=n(n-m)+m^2$) is an affine quotient group [27] of the group $Gr\subset GP\left(n\right)$ acting on the pencil of lines through A with the structure Equations (3), (4), (6) and (7);

5. 

the maximal quotient bundle is made from the quotient bundle of plane co-affine frames and the affine quotient bundle with the structure Equations (3)–(7).

Normalization of the space $\Pi$ is made by the fields of the following geometric patterns: the first kind normal, i.e., an $(n-m)$-plane $N_{n-m}$ intersecting the plane $P_m^0$ only at the point A and the second kind normal, i.e., an $(m-1)$-plane $N_{m-1}$ contained in the centered plane $P_m^0$ and not passing through its centre A (see, e.g., [11,28]).

Let us now analyze the dynamics of changes of the bundle $G(\Pi )$ at the consecutive canonizations:

• by placing the vertices $A_{\alpha }$ on the first normal $N_{n-m}$ (the 1st canonization);
• by placing the vertices $A_a$ on the second normal $N_{m-1}$ (the 2nd canonization);
• by simultaneous placing the vertices on the corresponding normals (full canonization).

Remark 4.

The space ${\Pi }^1$ or ${\Pi }^2$ is said to be a semi-normalized space in the first or second case, respectively, and the space ${\Pi }^{1,2}$ is a normalized space in the third case.

4.1. The Bundle $G^1(\Pi )$

We put the vertices $A_{\alpha }$ on the first normal $N_{n-m}$. Then, the following relations must hold:

$${\omega }_{\alpha }^a=g_{\alpha \beta }^a{\omega }^{\beta }+g_{\alpha b}^a{\omega }^b+g_{\alpha \beta }^{ab}{\omega }_b^{\beta },$$

(8)

with the differential congruences

$$\Delta g_{\alpha \beta }^a+g_{\alpha \beta }^{ab}{\omega }_b\equiv 0,\Delta g_{\alpha b}^a-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta g_{\alpha \beta }^{ab}\equiv 0(mod{\omega }^{\alpha },{\omega }_a^{\alpha },{\omega }^a).$$

(9)

Here, and subsequently, the differential operator $\Delta$ acts in the standard way (see, e.g., [29])

$$\Delta g_{\alpha \beta }^a=d{g}_{\alpha \beta }^a+g_{\alpha \beta }^c{\omega }_c^a-g_{\gamma \beta }^a{\omega }_{\alpha }^{\gamma }-g_{\alpha \gamma }^a{\omega }_{\beta }^{\gamma }.$$

Taking into account (8), from the structure Equation (2), we have

$$\begin{array}{c}D{\omega }_b^a={\omega }_b^c\wedge {\omega }_c^a+({\delta }_c^a{\omega }_b+{\delta }_b^a{\omega }_c)\wedge {\omega }^c+{\delta }_b^a{\omega }_{\alpha }\wedge {\omega }^{\alpha }+\\ {(\dots )}_{\alpha \beta }^a{\omega }_b^{\alpha }\wedge {\omega }^{\beta }+{(\dots )}_{\alpha c}^a{\omega }_b^{\alpha }\wedge {\omega }^c+{(\dots )}_{\alpha \beta }^{ac}{\omega }_b^{\alpha }\wedge {\omega }_c^{\beta };\end{array}$$

(10)

$$D{\omega }_a={\omega }_a^b\wedge {\omega }_b-{\omega }_{\alpha }\wedge {\omega }_a^{\alpha };$$

(11)

$$\begin{array}{c}D{\omega }_{\beta }^{\alpha }={\omega }_{\beta }^{\gamma }\wedge {\omega }_{\gamma }^{\alpha }+{\delta }_{\beta }^{\alpha }{\omega }_a\wedge {\omega }^a+({\delta }_{\gamma }^{\alpha }{\omega }_{\beta }+{\delta }_{\beta }^{\alpha }{\omega }_{\gamma })\wedge {\omega }^{\gamma }+\\ {(\dots )}_{\beta \gamma }^a{\omega }^{\gamma }\wedge {\omega }_a^{\alpha }+{(\dots )}_{\beta b}^a{\omega }^b\wedge {\omega }_a^{\alpha }+{(\dots )}_{\beta \gamma }^{ab}{\omega }_b^{\gamma }\wedge {\omega }_a^{\alpha };\end{array}$$

(12)

$$D{\omega }_{\alpha }={\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }+{\omega }^{\beta }\wedge g_{\alpha \beta }^a{\omega }_a+{\omega }^b\wedge g_{\alpha b}^a{\omega }_a+{\omega }_b^{\beta }\wedge g_{\alpha \beta }^{ab}{\omega }_a.$$

(13)

From Equations (10)–(13), it can be argued that, at the first canonization, the principal bundle $G(\Pi )$ is narrowed to the principal bundle $G^1(\Pi )$; its typical fiber is the stationary subgroup $G^1\subset G$ of a pair of the affine additional planes $\{P_m,N_{n-m}\}$. There are four quotient bundles in the subbundle $G^1(\Pi )$:

• the quotient bundle of plane linear frames with the structure Equations (3) and (10);
• the quotient bundle of normal linear frames (3) and (12);
• the quotient bundle of plane co-affine frames (3), (10), and (11);
• the quotient bundle of normal co-affine frames (3), (12), and (13).

4.2. The Bundle $G^2(\Pi )$

If we do not use the previous canonization and place the vertices $A_a$ on the second normal $N_{m-1}$, then

$${\omega }_a=g_{a\alpha }{\omega }^{\alpha }+g_{ab}{\omega }^b+g_{a\alpha }^b{\omega }_b^{\alpha }$$

(14)

and

$$\Delta g_{a\alpha }-g_{ab}{\omega }_{\alpha }^b\equiv 0,\Delta g_{ab}\equiv 0,\Delta g_{a\alpha }^b+{\delta }_a^b{\omega }_{\alpha }\equiv 0(mod{\omega }^{\alpha },{\omega }_a^{\alpha },{\omega }^a).$$

(15)

Then, from the structure Equation (2), we have

$$\begin{array}{c}D{\omega }_b^a={\omega }_b^c\wedge {\omega }_c^a+{\omega }_b^{\alpha }\wedge {\omega }_{\alpha }^a+{\delta }_b^a{\omega }_{\alpha }\wedge {\omega }^{\alpha }+\\ {(\dots )}_{b\alpha c}^a{\omega }^{\alpha }\wedge {\omega }^c+{(\dots )}_{bce}^a{\omega }^c\wedge {\omega }^e+{(\dots )}_{bc\alpha }^{ae}{\omega }_e^{\alpha }\wedge {\omega }^c;\end{array}$$

(16)

$$\begin{array}{c}D{\omega }_{\beta }^{\alpha }={\omega }_{\beta }^{\gamma }\wedge {\omega }_{\gamma }^{\alpha }+({\delta }_{\gamma }^{\alpha }{\omega }_{\beta }+{\delta }_{\beta }^{\alpha }{\omega }_{\gamma })\wedge {\omega }^{\gamma }-{\omega }_a^{\alpha }\wedge {\omega }_{\beta }^a+\\ {(\dots )}_{\beta \gamma a}^{\alpha }{\omega }^{\gamma }\wedge {\omega }^a+{(\dots )}_{\beta ba}^{\alpha }{\omega }^b\wedge {\omega }^a+{(\dots )}_{\beta \gamma a}^{\alpha b}{\omega }_b^{\gamma }\wedge {\omega }^a;\end{array}$$

(17)

$$D{\omega }_{\alpha }^a={\omega }_{\alpha }^b\wedge {\omega }_b^a+{\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }^a+{\omega }_{\alpha }\wedge {\omega }^a;$$

(18)

$$D{\omega }_{\alpha }={\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }-{\omega }^{\beta }\wedge g_{a\beta }{\omega }_{\alpha }^a-{\omega }^b\wedge g_{ab}{\omega }_{\alpha }^a-{\omega }_b^{\beta }\wedge g_{a\beta }^b{\omega }_{\alpha }^a.$$

(19)

In fact, according to Equations (16)–(19), we can make a conclusion that, at the second canonization, the principal bundle $G(\Pi )$ is narrowed to the principal bundle $G^2(\Pi )$; its typical fiber is the stationary subgroup $G^2\subset G$ of the pair $\{A,N_{m-1}\}$. There are four quotient bundles in the subbundle $G^2(\Pi )$:

• the quotient bundle of plane linear frames with the structure Equations (3) and (16);
• the quotient bundle of normal linear frames (3) and (17);
• the bundle $H(\Pi )$ (3), (16)–(18) whose typical fiber H is an affine quotient group of $G_2$;
• the bundle of normal co-affine frames (3), (17), and (19).

4.3. The Bundle $G^{1,2}(\Pi )$

Now, suppose that we have already made canonizations considered in items 4.1 and 4.2 simultaneously, that is, $A_{\alpha }\in N_{n-m}$ and $A_a\in N_{m-1}$. In this case, conditions (8) and (14) are satisfied and the structure Equation (2) will become

$$\begin{array}{c}D{\omega }_b^a={\omega }_b^c\wedge {\omega }_c^a+{\delta }_b^a{\omega }_{\alpha }\wedge {\omega }^{\alpha }+{(\dots )}_{bc\alpha }^a{\omega }^{\alpha }\wedge {\omega }^c+{(\dots )}_{bce}^a{\omega }^c\wedge {\omega }^e+\\ {(\dots )}_{b\alpha c}^{ae}{\omega }_e^{\alpha }\wedge {\omega }^c+{(\dots )}_{\alpha \beta }^a{\omega }_b^{\alpha }\wedge {\omega }^{\beta }+{(\dots )}_{\alpha \beta }^{ac}{\omega }_b^{\alpha }\wedge {\omega }_c^{\beta };\end{array}$$

(20)

$$\begin{array}{c}D{\omega }_{\beta }^{\alpha }={\omega }_{\beta }^{\gamma }\wedge {\omega }_{\gamma }^{\alpha }+({\delta }_{\gamma }^{\alpha }{\omega }_{\beta }+{\delta }_{\beta }^{\alpha }{\omega }_{\gamma })\wedge {\omega }^{\gamma }+{(\dots )}_{\beta \gamma a}^{\alpha }{\omega }^{\gamma }\wedge {\omega }^a+\\ +{(\dots )}_{\beta ba}^{\alpha }{\omega }^b\wedge {\omega }^a+{(\dots )}_{\beta \gamma a}^{\alpha b}{\omega }_b^{\gamma }\wedge {\omega }^a+{(\dots )}_{\beta \gamma }^a{\omega }^{\gamma }\wedge {\omega }_a^{\alpha }+{(\dots )}_{\beta \gamma }^{ab}{\omega }_b^{\gamma }\wedge {\omega }_a^{\alpha };\end{array}$$

(21)

$$\begin{array}{c}D{\omega }_{\alpha }={\omega }_{\alpha }^{\beta }\wedge {\omega }_{\beta }+{(\dots )}_{\alpha \beta \gamma }{\omega }^{\beta }\wedge {\omega }^{\gamma }+{(\dots )}_{\alpha \beta b}{\omega }^{\beta }\wedge {\omega }^b+{(\dots )}_{\alpha \beta \gamma }^b{\omega }^{\beta }\wedge {\omega }_b^{\gamma }+\\ {(\dots )}_{\alpha bc}{\omega }^b\wedge {\omega }^c+{(\dots )}_{\alpha b\beta }^c{\omega }^b\wedge {\omega }_c^{\beta }+{(\dots )}_{\alpha \beta \gamma }^{bc}{\omega }_b^{\beta }\wedge {\omega }_c^{\gamma }.\end{array}$$

(22)

Obviously, $G^{1,2}$ is the stationary subgroup of the centered $(n-m)$ pair $\{N_{n-m}^{*}$, $N_{m-1}\}$ [30]—then at the full canonization from narrowing $G^{1,2}(\Pi )$ of the principal bundle $G(\Pi )$. The following three quotient bundles are allocated:

• the bundle of plane linear frames (3) and (20);
• the bundle of normal linear frames (3) and (21);
• the bundle of normal co-affine frames (3), (21), and (22).

5. A Connection on the Bundle Associated with the Space $\mathbf{\Pi }$

Using the Laptev–Lumiste method (see, e.g., [4,31]), on the principal bundle $G(\Pi )$, we define a fundamental-group connection by the forms

$$\begin{array}{c}{\tilde{\omega }}_b^a={\omega }_b^a-L_{b\alpha }^a{\omega }^{\alpha }-L_{bc}^a{\omega }^c-{\Gamma }_{b\alpha }^{ac}{\omega }_c^{\alpha },{\tilde{\omega }}_{\beta }^{\alpha }={\omega }_{\beta }^{\alpha }-L_{\beta \gamma }^{\alpha }{\omega }^{\gamma }-L_{\beta a}^{\alpha }{\omega }^a-{\Gamma }_{\beta \gamma }^{\alpha a}{\omega }_a^{\gamma },\\ {\tilde{\omega }}_{\alpha }^a={\omega }_{\alpha }^a-L_{\alpha \beta }^a{\omega }^{\beta }-L_{\alpha b}^a{\omega }^b-{\Gamma }_{\alpha \beta }^{ab}{\omega }_b^{\beta },{\tilde{\omega }}_a={\omega }_a-{\Gamma }_{a\alpha }{\omega }^{\alpha }-{\Gamma }_{ab}{\omega }^b-{\Pi }_{a\alpha }^b{\omega }_b^{\alpha },\\ {\tilde{\omega }}_{\alpha }={\omega }_{\alpha }-{\Gamma }_{\alpha \beta }{\omega }^{\beta }-{\Gamma }_{\alpha a}{\omega }^a-{\Pi }_{\alpha \beta }^a{\omega }_a^{\beta }.\end{array}$$

(23)

The components of the connection object [4]

$$\Gamma =\{L_{b\alpha }^a,L_{bc}^a,{\Gamma }_{b\alpha }^{ac},L_{\beta \gamma }^{\alpha },L_{\beta a}^{\alpha },{\Gamma }_{\beta \gamma }^{\alpha a},L_{\alpha \beta }^a,L_{\alpha b}^a,{\Gamma }_{\alpha \beta }^{ab},{\Gamma }_{a\alpha },{\Gamma }_{ab},{\Pi }_{a\alpha }^b,{\Gamma }_{\alpha \beta },{\Gamma }_{\alpha a},{\Pi }_{\alpha \beta }^a\}$$

satisfy the following differential congruences modulo the basis forms ${\omega }^{\alpha }$, ${\omega }_a^{\alpha }$, ${\omega }^a$:

$$\begin{array}{c}\Delta L_{b\alpha }^a-L_{bc}^a{\omega }_{\alpha }^c+{\Gamma }_{b\alpha }^{ac}{\omega }_c-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta L_{bc}^a-{\delta }_c^a{\omega }_b-{\delta }_b^a{\omega }_c\equiv 0,\Delta {\Gamma }_{b\alpha }^{ac}+{\delta }_b^c{\omega }_{\alpha }^a\equiv 0,\\ \Delta L_{\beta \gamma }^{\alpha }-L_{\beta a}^{\alpha }{\omega }_{\gamma }^a+{\Gamma }_{\beta \gamma }^{\alpha a}{\omega }_a-{\delta }_{\beta }^{\alpha }{\omega }_{\gamma }-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }\equiv 0,\Delta L_{\beta a}^{\alpha }-{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta {\Gamma }_{\beta \gamma }^{\alpha a}-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }^a\equiv 0,\\ \Delta L_{\alpha \beta }^a+{\Gamma }_{\alpha \beta }^{ab}{\omega }_b-L_{b\beta }^a{\omega }_{\alpha }^b+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }^a-L_{\alpha b}^a{\omega }_{\beta }^b\equiv 0,\\ \Delta L_{\alpha b}^a-L_{cb}^a{\omega }_{\alpha }^c+L_{\alpha b}^{\beta }{\omega }_{\beta }^a-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta {\Gamma }_{\alpha \beta }^{ab}-{\Gamma }_{d\beta }^{ab}{\omega }_{\alpha }^d+{\Gamma }_{\alpha \beta }^{\gamma b}{\omega }_{\gamma }^a\equiv 0,\\ \Delta {\Gamma }_{a\alpha }-{\Gamma }_{ab}{\omega }_{\alpha }^b+({\Pi }_{a\alpha }^b+L_{a\alpha }^b){\omega }_b\equiv 0,\\ \Delta {\Gamma }_{ab}+L_{ab}^c{\omega }_c\equiv 0,\Delta {\Pi }_{a\alpha }^b+{\Gamma }_{a\alpha }^{db}{\omega }_d+{\delta }_a^b{\omega }_{\alpha }\equiv 0,\\ \Delta {\Gamma }_{\alpha \beta }-{\Gamma }_{\alpha a}{\omega }_{\beta }^a+({\Pi }_{\alpha \beta }^a+L_{\alpha \beta }^a){\omega }_a-{\Gamma }_{a\beta }{\omega }_{\alpha }^a+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }\equiv 0,\\ \Delta {\Gamma }_{\alpha a}-{\Gamma }_{ba}{\omega }_{\alpha }^b+L_{\alpha a}^b{\omega }_b+L_{\alpha a}^{\beta }{\omega }_{\beta }\equiv 0,\Delta {\Pi }_{\alpha \beta }^a+{\Gamma }_{\alpha \beta }^{ba}{\omega }_b-{\Pi }_{c\beta }^a{\omega }_{\alpha }^c+{\Gamma }_{\alpha \beta }^{\gamma a}{\omega }_{\gamma }\equiv 0.\end{array}$$

(24)

Remark 5.

The connection object Γ contains the following five geometric subobjects ${\Gamma }_1=\{L_{b\alpha }^a, L_{bc}^a,{\Gamma }_{b\alpha }^{ac}\}$, ${\Gamma }_2=\{L_{\beta \gamma }^{\alpha }, L_{\beta a}^{\alpha }, {\Gamma }_{\beta \gamma }^{\alpha a}\}$, ${\Gamma }_3=\{{\Gamma }_1, {\Gamma }_{a\alpha }, {\Gamma }_{ab},{\Pi }_{a\alpha }^b\}$, ${\Gamma }_4=\{{\Gamma }_1, {\Gamma }_2, L_{\alpha \beta }^a, L_{\alpha b}^a, {\Gamma }_{\alpha \beta }^{ab}\}$, and ${\Gamma }_5=\{{\Gamma }_3\backslash {\Gamma }_1, {\Gamma }_4\}$. These subobjects determine connections on the corresponding (see Remark 3) quotient bundles.

Let us consider the dynamics of changes of the connection $\Gamma$ at consecutive canonizations and we will be convinced that the connection $\Gamma$ is not uniquely induced at the normalization of the space $\Pi$.

5.1. The Connection Object at Adaptation of the Moving Frame to the First Normal

By placing the vertices $A_{\alpha }$ on the first normal $N_{n-m}$, condition (8) is satisfied, that is, the forms ${\omega }_{\alpha }^a$ become principal and the connection object $\Gamma$ is narrowed to the object ${\Gamma }^1=\Gamma \backslash ({\Gamma }_4\backslash ({\Gamma }_1\cup {\Gamma }_2))$; and differential congruences for its components have the form

$$\begin{array}{c}\Delta L_{b\alpha }^a+{\Gamma }_{b\alpha }^{ac}{\omega }_c-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta L_{bc}^a-{\delta }_c^a{\omega }_b-{\delta }_b^a{\omega }_c\equiv 0,\Delta {\Gamma }_{b\alpha }^{ac}\equiv 0,\\ \Delta L_{\beta \gamma }^{\alpha }+{\Gamma }_{\beta \gamma }^{\alpha a}{\omega }_a-{\delta }_{\beta }^{\alpha }{\omega }_{\gamma }-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }\equiv 0,\Delta L_{\beta a}^{\alpha }-{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta {\Gamma }_{\beta \gamma }^{\alpha a}\equiv 0,\\ \Delta {\Gamma }_{a\alpha }+({\Pi }_{a\alpha }^b+L_{a\alpha }^b){\omega }_b\equiv 0,\Delta {\Gamma }_{ab}+L_{ab}^c{\omega }_c\equiv 0,\\ \Delta {\Pi }_{a\alpha }^b+{\Gamma }_{a\alpha }^{db}{\omega }_d+{\delta }_a^b{\omega }_{\alpha }\equiv 0,\Delta {\Gamma }_{\alpha \beta }+({\Pi }_{\alpha \beta }^a+g_{\alpha \beta }^a){\omega }_a+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }\equiv 0,\\ \Delta {\Gamma }_{\alpha a}+L_{\alpha a}^{\beta }{\omega }_{\beta }+g_{\alpha a}^b{\omega }_b\equiv 0,\Delta {\Pi }_{\alpha \beta }^a+{\Gamma }_{\alpha \beta }^{\gamma a}{\omega }_{\gamma }+g_{\alpha \beta }^{ba}{\omega }_b\equiv 0.\end{array}$$

(25)

All of this points to the fact that the following theorem holds.

Theorem 1.

At an adaptation of the moving frame to a field of the first normals the connection object Γ is reduced to the object ${\Gamma }^1$. The object ${\Gamma }^1$ contains three subobjects ${\Gamma }_1^1$, ${\Gamma }_2^1$, ${\Gamma }_3^1$ that set connections on the quotient bundles of plane linear frames, normal linear frames, and plane co-affine frames, respectively.

5.2. Connection Object at Adaptation of the Moving Frame to the Second Normal

Without using the previous canonization and placing the vertices $A_a$ on the second normal $N_{m-1}$, we get condition (14). The connection object $\Gamma$ is narrowed to the object ${\Gamma }^2=\Gamma \backslash ({\Gamma }_3\backslash {\Gamma }_1)$ with the following congruences for its components:

$$\begin{array}{c}\Delta L_{b\alpha }^a-L_{bc}^a{\omega }_{\alpha }^c-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta L_{bc}^a\equiv 0,\Delta {\Gamma }_{b\alpha }^{ac}+{\delta }_b^c{\omega }_{\alpha }^a\equiv 0,\\ \Delta L_{\beta \gamma }^{\alpha }-L_{\beta a}^{\alpha }{\omega }_{\gamma }^a-{\delta }_{\beta }^{\alpha }{\omega }_{\gamma }-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }\equiv 0,\Delta L_{\beta a}^{\alpha }\equiv 0,\Delta {\Gamma }_{\beta \gamma }^{\alpha a}-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }^a\equiv 0,\\ \Delta L_{\alpha \beta }^a-L_{b\beta }^a{\omega }_{\alpha }^b-L_{\alpha b}^a{\omega }_{\beta }^b+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }^a\equiv 0,\Delta L_{\alpha b}^a-L_{cb}^a{\omega }_{\alpha }^c+L_{\alpha b}^{\beta }{\omega }_{\beta }^a-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\\ \Delta {\Gamma }_{\alpha \beta }^{ab}-{\Gamma }_{d\beta }^{ab}{\omega }_{\alpha }^d+{\Gamma }_{\alpha \beta }^{\gamma b}{\omega }_{\gamma }^a\equiv 0,\Delta {\Gamma }_{\alpha \beta }-{\Gamma }_{\alpha a}{\omega }_{\beta }^a+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }-g_{a\beta }{\omega }_{\alpha }^a\equiv 0,\\ \Delta {\Gamma }_{\alpha a}+L_{\alpha a}^{\beta }{\omega }_{\beta }-g_{ba}{\omega }_{\alpha }^b\equiv 0,\Delta {\Pi }_{\alpha \beta }^a+{\Gamma }_{\alpha \beta }^{\gamma a}{\omega }_{\gamma }-g_{b\beta }^a{\omega }_{\alpha }^b\equiv 0.\end{array}$$

(26)

The arguments given above prove Theorem 2.

Theorem 2.

At an adaptation of the moving frame to a field of the second normals, the connection object Γ is reduced to the object ${\Gamma }^2$. The object ${\Gamma }^2$ contains three subobjects ${\Gamma }_1^2$, ${\Gamma }_2^2$, ${\Gamma }_3^2$ that set connections on the quotient bundles of plane linear frames, normal linear frames, and affine quotient bundle, respectively.

5.3. Connection Object at Normalization

By making both canonizations simultaneously, that is, placing the vertices $A_{\alpha }$ on the first normal $N_{n-m}$ and the vertices $A_a$ on the second normal $N_{m-1}$, the differential congruences for object’s ${\Gamma }^{1,2}=\Gamma \backslash (({\Gamma }_4\backslash ({\Gamma }_1\cup {\Gamma }_2))\cup ({\Gamma }_3\backslash {\Gamma }_1))$ components will become

$$\begin{array}{c}\Delta L_{b\alpha }^a-{\delta }_b^a{\omega }_{\alpha }\equiv 0,\Delta L_{bc}^a\equiv 0,\Delta {\Gamma }_{b\alpha }^{ac}\equiv 0,\Delta L_{\beta \gamma }^{\alpha }-{\delta }_{\beta }^{\alpha }{\omega }_{\gamma }-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }\equiv 0,\Delta L_{\beta a}^{\alpha }\equiv 0,\\ \Delta {\Gamma }_{\beta \gamma }^{\alpha a}\equiv 0,\Delta {\Gamma }_{\alpha \beta }+L_{\alpha \beta }^{\gamma }{\omega }_{\gamma }\equiv 0,\Delta {\Gamma }_{\alpha a}+L_{\alpha a}^{\beta }{\omega }_{\beta }\equiv 0,\Delta {\Pi }_{\alpha \beta }^a+{\Gamma }_{\alpha \beta }^{\gamma a}{\omega }_{\gamma }\equiv 0.\end{array}$$

(27)

In addition, we have the following theorem.

Theorem 3.

At an adaptation of the moving frame to normalization of the space Π, the connection object Γ is reduced to the object ${\Gamma }^{1,2}$. The object ${\Gamma }^{1,2}$ contains two subobjects ${\Gamma }_1^{1,2}$, ${\Gamma }_2^{1,2}$ that set connections on the quotient bundles of plane and normal linear frames.

5.4. Reduced Connection Objects

With the help of conditions (8), the forms ${\omega }_{\alpha }^a$ become principal, and, therefore, congruences for the components $L_{\alpha \beta }^a$, $L_{\alpha b}^a$, ${\Gamma }_{\alpha \beta }^{ab}$ in (24) will be carried out identically and they can be omitted. Using conditions (8) in the rest of the differential congruences (24), the components of the reduced connection object ${\Gamma }^I$ will satisfy (25) if the following conditions hold:

$$g_{\alpha \beta }^a=L_{\alpha \beta }^a,g_{\alpha a}^b=L_{\alpha a}^b,g_{\alpha \beta }^{ab}={\Gamma }_{\alpha \beta }^{ab}.$$

(28)

Theorem 4.

The reduced object ${\Gamma }^I$ coincides with the object ${\Gamma }^1$ only if conditions (28) hold, where the object ${\Gamma }^1$ gives a connection on the reduced bundle that arises at the adaptation of the moving frame to a field of the first normals.

By substituting conditions (14) into the differential congruences for components of the connection object $\Gamma$, we get congruences (26) with conditions

$$g_{a\beta }={\Gamma }_{a\beta },g_{ab}={\Gamma }_{ab},g_{b\alpha }^a={\Pi }_{b\alpha }^a$$

(29)

for the components of the reduced connection object ${\Gamma }^{II}$.

Theorem 5.

The reduced object ${\Gamma }^{II}$ coincides with the object ${\Gamma }^2$ only in the case (29), where the object ${\Gamma }^2$ gives a connection on the reduced bundle that arises at the adaptation of the moving frame to a field of the second normals.

Taking into account conditions (8) and (14) in the first six and last three differential congruences (24), we have that the components of the reduced connection object ${\Gamma }^{I,II}$ satisfy the differential congruences (27).

Theorem 6.

The reduced connection object ${\Gamma }^{I,II}$ coincides with the object ${\Gamma }^{1,2}$, which gives a connection on the reduced bundle at the adaptation of the moving frame to the normalization of the space Π.

Remark 6.

Adaptations of the moving frame cause the reductions of associated bundle and differential congruences for components of group connection object. Semi-canonizations (the first and second canonizations) lead to reductions of bundle and connection object, but, according to Theorems 4 and 5, the reduced connection objects can differ from the objects specifying connections on the reduced bundles.

6. Curvature and Torsion Objects

Let us now consider curvature and torsion objects [4] of group connection on the bundle associated with the space $\Pi$ of centered planes at a transition to the normalized space ${\Pi }^{1,2}$.

6.1. Curvature Objects

Generic curvature R of the connection $\Gamma$ of the space $\Pi$ was studied in [22]. Denote by $R^1$ and $R^2$ the objects of reduced curvature on the bundle associated with the space $\Pi$.

6.1.1. Curvature Object at the First Canonization

At the adaptation of the moving frame to a field of the first normals, we have

$$\begin{array}{c}D{\tilde{\omega }}_a={\tilde{\omega }}_a^b\wedge {\tilde{\omega }}_b+R_{a\alpha \beta }{\omega }^{\alpha }\wedge {\omega }^{\beta }+R_{abc}{\omega }^b\wedge {\omega }^c+R_{a\alpha b}{\omega }^{\alpha }\wedge {\omega }^b+\\ K_{a\alpha \beta }^b{\omega }^{\alpha }\wedge {\omega }_b^{\beta }+R_{ab\alpha }^c{\omega }^b\wedge {\omega }_c^{\alpha }+K_{a\alpha \beta }^{bc}{\omega }_b^{\alpha }\wedge {\omega }_c^{\beta },\\ D{\tilde{\omega }}_b^a={\tilde{\omega }}_b^c\wedge {\tilde{\omega }}_c^a+R_{b\alpha \beta }^a{\omega }^{\alpha }\wedge {\omega }^{\beta }+R_{bce}^a{\omega }^c\wedge {\omega }^e+R_{b\alpha c}^a{\omega }^{\alpha }\wedge {\omega }^c+\\ R_{b\alpha \beta }^{ac}{\omega }^{\alpha }\wedge {\omega }_c^{\beta }+R_{bc\alpha }^{ad}{\omega }^c\wedge {\omega }_d^{\alpha }+R_{b\alpha \beta }^{acd}{\omega }_c^{\alpha }\wedge {\omega }_d^{\beta },\\ D{\tilde{\omega }}_{\beta }^{\alpha }={\tilde{\omega }}_{\beta }^{\gamma }\wedge {\tilde{\omega }}_{\gamma }^{\alpha }+R_{\beta \gamma \mu }^{\alpha }{\omega }^{\gamma }\wedge {\omega }^{\mu }+R_{\beta ab}^{\alpha }{\omega }^a\wedge {\omega }^b+R_{\beta \gamma a}^{\alpha }{\omega }^{\gamma }\wedge {\omega }^a+\\ R_{\beta \gamma \mu }^{\alpha a}{\omega }^{\gamma }\wedge {\omega }_a^{\mu }+R_{\beta a\gamma }^{\alpha b}{\omega }^a\wedge {\omega }_b^{\gamma }+R_{\beta \gamma \mu }^{\alpha ab}{\omega }_a^{\gamma }\wedge {\omega }_b^{\mu },\end{array}$$

$$\begin{array}{c}D{\tilde{\omega }}_{\alpha }={\tilde{\omega }}_{\alpha }^a\wedge {\tilde{\omega }}_a+{\tilde{\omega }}_{\alpha }^{\beta }\wedge {\tilde{\omega }}_{\beta }+R_{\alpha \beta \gamma }{\omega }^{\beta }\wedge {\omega }^{\gamma }+R_{\alpha ab}{\omega }^a\wedge {\omega }^b+R_{\alpha \beta a}{\omega }^{\beta }\wedge {\omega }^a+\\ K_{\alpha \beta \gamma }^a{\omega }^{\beta }\wedge {\omega }_a^{\gamma }+R_{\alpha a\beta }^b{\omega }^a\wedge {\omega }_b^{\beta }+K_{\alpha \beta \gamma }^{ab}{\omega }_a^{\beta }\wedge {\omega }_b^{\gamma }.\end{array}$$

The components of the curvature object R are determined by the following relations:

$$\begin{array}{c}{R}_{a\alpha \beta }={\Gamma }_{a\left[\alpha \beta \right]}-{\Gamma }_{ab}{g}_{\left[\alpha \beta \right]}^b-L_{a[\alpha }^b{\Gamma }_{b\beta ]},R_{a\alpha b}={\Gamma }_{a\alpha b}-{\Gamma }_{ab\alpha }-{\Gamma }_{ac}{g}_{\alpha b}^c-L_{a\alpha }^c{\Gamma }_{cb}+L_{ab}^c{\Gamma }_{c\alpha },\\ R_{abc}={\Gamma }_{a\left[bc\right]}-L_{a[b}^e{\Gamma }_{ec]},K_{a\alpha \beta }^b={\Gamma }_{a\alpha \beta }^b-{\Pi }_{a\beta \alpha }^b-{\Gamma }_{ac}{g}_{\alpha \beta }^{cb}-L_{a\alpha }^c{\Pi }_{c\beta }^b+{\Gamma }_{a\beta }^{cb}{\Gamma }_{c\alpha },\\ R_{ab\alpha }^c={\Gamma }_{ab\alpha }^c-{\Pi }_{a\alpha b}^c+{\Gamma }_{a\alpha }^{ec}{\Gamma }_{eb}-L_{ab}^e{\Pi }_{e\alpha }^c-{\delta }_b^c{\Gamma }_{a\alpha },K_{a\alpha \beta }^{bc}={\Pi }_a{[}_{\alpha \beta }^{bc}]-{\Gamma }_a^e{[}_{\alpha }^b{\Pi }_{e\beta }^c],\\ R_{b\alpha \beta }^a=L_{b\left[\alpha \beta \right]}^a-L_{b[\alpha }^c{L}_{c\beta ]}^a-L_{bc}^a{g}_{\left[\alpha \beta \right]}^c,R_{bce}^a=L_{b\left[ce\right]}^a-L_{b[c}^d{L}_{de]}^a,\\ R_{b\alpha c}^a=L_{b\alpha c}^a-L_{bc\alpha }^a-L_{b\alpha }^e{L}_{ec}^a+L_{bc}^e{L}_{e\alpha }^a-L_{be}^a{g}_{\alpha c}^e,\end{array}$$

$$\begin{array}{c}{R}_{b\alpha \beta }^{ac}=L_{b\alpha \beta }^{ac}-{\Gamma }_{b\beta \alpha }^{ac}-L_{b\alpha }^e{\Gamma }_{e\beta }^{ac}+{\Gamma }_{b\beta }^{ec}{L}_{e\alpha }^a-L_{be}^a{g}_{\alpha \beta }^{ec}-{\delta }_b^c{g}_{\beta \alpha }^a,\\ R_{bc\alpha }^{ad}=L_{bc\alpha }^{ad}-{\Gamma }_{b\alpha c}^{ad}-{\delta }_c^d{L}_{b\alpha }^a-L_{bc}^e{\Gamma }_{e\alpha }^{ad}+{\Gamma }_{b\alpha }^{ed}{L}_{ec}^a-{\delta }_b^d{g}_{\alpha c}^a,\\ R_{b\alpha \beta }^{acd}={\Gamma }_b^a{[}_{\alpha \beta }^{cd}]-{\Gamma }_b^e{[}_{\alpha }^c{\Gamma }_{e\beta }^{ad}]+{\delta }_b^{\lceil c}{g}_{\lfloor \alpha \beta }^{ad}],\\ R_{\beta \gamma \mu }^{\alpha }=L_{\beta \left[\gamma \mu \right]}^{\alpha }-L_{\beta [\gamma }^{\eta }{L}_{\eta \mu ]}^{\alpha }-L_{\beta a}^{\alpha }{g}_{\left[\gamma \mu \right]}^a,R_{\beta ab}^{\alpha }=L_{\beta \left[ab\right]}^{\alpha }-L_{\beta [a}^{\gamma }{L}_{\gamma b]}^{\alpha },\\ R_{\beta \gamma a}^{\alpha }=L_{\beta \gamma a}^{\alpha }-L_{\beta a\gamma }^{\alpha }-L_{\beta \gamma }^{\mu }{L}_{\mu a}^{\alpha }+L_{\beta a}^{\mu }{L}_{\mu \gamma }^{\alpha }-L_{\beta b}^{\alpha }{g}_{\gamma a}^b,\\ R_{\beta \gamma \mu }^{\alpha a}=L_{\beta \gamma \mu }^{\alpha a}-{\Gamma }_{\beta \mu \gamma }^{\alpha a}-L_{\beta \gamma }^{\eta }{\Gamma }_{\eta \mu }^{\alpha a}+{\Gamma }_{\beta \mu }^{\eta a}{L}_{\eta \gamma }^{\alpha }-L_{\beta b}^{\alpha }{g}_{\gamma \mu }^{ba}+{\delta }_{\mu }^{\alpha }{g}_{\beta \gamma }^a,\\ R_{\beta a\gamma }^{\alpha b}=L_{\beta a\gamma }^{\alpha b}-{\Gamma }_{\beta \gamma a}^{\alpha b}-L_{\beta a}^{\mu }{\Gamma }_{\mu \gamma }^{\alpha b}+{\Gamma }_{\beta \gamma }^{\mu b}{L}_{\mu a}^{\alpha }-{\delta }_a^b{L}_{\beta \gamma }^{\alpha }+{\delta }_{\gamma }^{\alpha }{g}_{\beta a}^b,\\ R_{\beta \gamma \mu }^{\alpha ab}={\Gamma }_{\beta }^{\alpha }{[}_{\gamma \mu }^{ab}]-{\Gamma }_{\beta }^{\eta }{[}_{\gamma }^a{\Gamma }_{\eta \mu }^{\alpha b}]-{\delta }_{\lfloor \gamma }^{\alpha }{g}_{\beta \mu }^{\lceil ab}],R_{\alpha \beta \gamma }={\Gamma }_{\alpha \left[\beta \gamma \right]}-L_{\alpha [\beta }^{\mu }{\Gamma }_{\mu \gamma ]}-{\Gamma }_{\alpha a}{g}_{\left[\beta \gamma \right]}^a,R_{\alpha ab}={\Gamma }_{\alpha \left[ab\right]}-L_{\alpha [a}^{\beta }{\Gamma }_{\beta b]},\\ R_{\alpha \beta a}={\Gamma }_{\alpha \beta a}-{\Gamma }_{\alpha a\beta }-L_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma a}+L_{\alpha a}^{\gamma }{\Gamma }_{\gamma \beta }-{\Gamma }_{\alpha b}{g}_{\beta a}^b,\\ K_{\alpha \beta \gamma }^a={\Gamma }_{\alpha \beta \gamma }^a-{\Pi }_{\alpha \gamma \beta }^a-L_{\alpha \beta }^{\mu }{\Pi }_{\mu \gamma }^a+{\Gamma }_{\alpha \gamma }^{\mu a}{\Gamma }_{\mu \beta }-{\Gamma }_{\alpha b}{g}_{\beta \gamma }^{ba},\\ R_{\alpha a\beta }^b={\Gamma }_{\alpha a\beta }^b-{\Pi }_{\alpha \beta a}^b-L_{\alpha a}^{\gamma }{\Pi }_{\gamma \beta }^b+{\Gamma }_{\alpha \beta }^{\gamma b}{\Gamma }_{\gamma a}-{\delta }_a^b{\Gamma }_{\alpha \beta },K_{\alpha \beta \gamma }^{ab}={\Pi }_{\alpha }{[}_{\beta \gamma }^{ab}]-{\Gamma }_{\alpha }^{\mu }{[}_{\beta }^a{\Pi }_{\mu \gamma }^b].\end{array}$$

Here, and in what follows, square brackets mean alternation over extreme indices.

Theorem 7.

The curvature R is reduced to the object $R^1=\{R_{a\alpha \beta }$, $R_{abc}$, $R_{a\alpha b}$, $K_{a\alpha \beta }^b$, $R_{ab\alpha }^c$, $K_{a\alpha \beta }^{bc}$, $R_{b\alpha \beta }^a$, $R_{bce}^a$, $R_{b\alpha c}^a$, $R_{b\alpha \beta }^{ac}$, $R_{bc\alpha }^{ad}$, $R_{b\alpha \beta }^{acd}$, $R_{\beta \gamma \mu }^{\alpha }$, $R_{\beta ab}^{\alpha }$, $R_{\beta \gamma a}^{\alpha }$, $R_{\beta \gamma \mu }^{\alpha a}$, $R_{\beta a\gamma }^{ab}$, $R_{\beta \gamma \mu }^{\alpha ab}$, $R_{\alpha \beta \gamma }$, $R_{\alpha ab}$, $R_{\alpha \beta a}$, $K_{\alpha \beta \gamma }^a$, $R_{\alpha a\beta }^b$, $K_{\alpha \beta \gamma }^{ab}\}$; and the reduced curvature object $R^1$ of the semi-normalized space ${\Pi }^1$ is a quasi-tensor together with the quasi-tensor $\{g_{\alpha \beta }^a$, $g_{\alpha b}^a$, $g_{\alpha \beta }^{ab}\}$. The object $R^1$ contains three subtensors that are curvature objects of subconnections on the bundles of plane linear frames, normal linear frames, and plane co-affine frames.

Proof of Theorem 7.

Extending the differential equations for components of the connection object $\Gamma$ and using (9), the differential congruences of curvature object components may be written as

$$\begin{array}{c}\Delta R_{a\alpha \beta }+K_{a\left[\alpha \beta \right]}^b{\omega }_b+R_{a\alpha \beta }^b{\omega }_b\equiv 0,\Delta R_{a\alpha b}-(R_{ab\alpha }^c-R_{a\alpha b}^c){\omega }_c\equiv 0,\Delta R_{abc}+R_{abc}^e{\omega }_e\equiv 0,\\ \Delta K_{a\alpha \beta }^b+(2K_{a\alpha \beta }^{cb}+R_{a\alpha \beta }^{cb}){\omega }_c\equiv 0,\Delta R_{ab\alpha }^c+R_{ab\alpha }^{ec}{\omega }_e\equiv 0,\Delta K_{a\alpha \beta }^{bc}+R_{a\alpha \beta }^{ebc}{\omega }_e\equiv 0,\\ \Delta R_{b\alpha \beta }^a+R_{b\left[\alpha \beta \right]}^{ac}{\omega }_c\equiv 0,\Delta R_{bce}^a\equiv 0,\Delta R_{b\alpha c}^a-R_{bc\alpha }^{ae}{\omega }_e\equiv 0,\Delta R_{b\alpha \beta }^{ac}+2R_{b\alpha \beta }^{aec}{\omega }_e\equiv 0,\\ \Delta R_{bc\alpha }^{ad}\equiv 0,\Delta R_{b\alpha \beta }^{ace}\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha }+R_{\beta \left[\gamma \mu \right]}^{\alpha a}{\omega }_a\equiv 0,\Delta R_{\beta ab}^{\alpha }\equiv 0,\Delta R_{\beta \gamma a}^{\alpha }-R_{\beta a\gamma }^{\alpha b}{\omega }_b\equiv 0,\\ \Delta R_{\beta \gamma \mu }^{\alpha a}-2R_{\beta \mu \gamma }^{\alpha ab}{\omega }_b\equiv 0,\Delta R_{\beta a\gamma }^{\alpha b}\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha ab}\equiv 0,\\ \Delta R_{\alpha \beta \gamma }+R_{\alpha \beta \gamma }^{\mu }{\omega }_{\mu }+(K_{\alpha \left[\beta \gamma \right]}^a+g_{\alpha \left[\beta \gamma \right]}^a-g_{\alpha b}^a{g}_{\left[\beta \gamma \right]}^b){\omega }_a\equiv 0,\Delta R_{\alpha ab}+R_{\alpha ab}^{\beta }{\omega }_{\beta }+g_{\alpha \left[ab\right]}^c{\omega }_c\equiv 0,\\ \Delta R_{\alpha \beta a}+(g_{\alpha \beta a}^b-g_{\alpha a\beta }^b-g_{\alpha c}^b{g}_{\beta a}^c-R_{\alpha a\beta }^b){\omega }_b+R_{\alpha \beta a}^{\gamma }{\omega }_{\gamma }\equiv 0,\end{array}$$

$$\begin{array}{c}\Delta K_{\alpha \beta \gamma }^a+(2K_{\alpha \beta \gamma }^{ba}+g_{\alpha \beta \gamma }^{ba}-g_{\alpha \gamma \beta }^{ba}-g_{\alpha c}^b{g}_{\beta \gamma }^{ca}){\omega }_b+R_{\alpha \beta \gamma }^{\mu a}{\omega }_{\mu }\equiv 0,\\ \Delta R_{\alpha a\beta }^b+R_{\alpha a\beta }^{\gamma b}{\omega }_{\gamma }+(g_{\alpha a\beta }^{cb}-g_{\alpha \beta a}^{cb}-{\delta }_a^b{g}_{\alpha \beta }^c){\omega }_c\equiv 0,\Delta K_{\alpha \beta \gamma }^{ab}+R_{\alpha \beta \gamma }^{\mu ab}{\omega }_{\mu }+g_{\alpha }^c{[}_{\beta \gamma }^{ab}]{\omega }_c\equiv 0,\end{array}$$

which proves the theorem.  □

6.1.2. Curvature Object at the Second Canonization

At the adaptation of the moving frame to a field of the second normals, we get

$$\begin{array}{c}D{\tilde{\omega }}_b^a={\tilde{\omega }}_b^c\wedge {\tilde{\omega }}_c^a+R_{b\alpha \beta }^a{\omega }^{\alpha }\wedge {\omega }^{\beta }+R_{bce}^a{\omega }^c\wedge {\omega }^e+R_{b\alpha c}^a{\omega }^{\alpha }\wedge {\omega }^c+\\ R_{b\alpha \beta }^{ac}{\omega }^{\alpha }\wedge {\omega }_c^{\beta }+R_{bc\alpha }^{ad}{\omega }^c\wedge {\omega }_d^{\alpha }+R_{b\alpha \beta }^{acd}{\omega }_c^{\alpha }\wedge {\omega }_d^{\beta },\\ D{\tilde{\omega }}_{\beta }^{\alpha }={\tilde{\omega }}_{\beta }^{\gamma }\wedge {\tilde{\omega }}_{\gamma }^{\alpha }+R_{\beta \gamma \mu }^{\alpha }{\omega }^{\gamma }\wedge {\omega }^{\mu }+R_{\beta ab}^{\alpha }{\omega }^a\wedge {\omega }^b+R_{\beta \gamma a}^{\alpha }{\omega }^{\gamma }\wedge {\omega }^a+\\ R_{\beta \gamma \mu }^{\alpha a}{\omega }^{\gamma }\wedge {\omega }_a^{\mu }+R_{\beta a\gamma }^{\alpha b}{\omega }^a\wedge {\omega }_b^{\gamma }+R_{\beta \gamma \mu }^{\alpha ab}{\omega }_a^{\gamma }\wedge {\omega }_b^{\mu },\\ D{\tilde{\omega }}_{\alpha }^a={\tilde{\omega }}_{\alpha }^b\wedge {\tilde{\omega }}_b^a+{\tilde{\omega }}_{\alpha }^{\beta }\wedge {\tilde{\omega }}_{\beta }^a+R_{\alpha \beta \gamma }^a{\omega }^{\beta }\wedge {\omega }^{\gamma }+R_{\alpha bc}^a{\omega }^b\wedge {\omega }^c+R_{\alpha \beta b}^a{\omega }^{\beta }\wedge {\omega }^b+\\ R_{\alpha b\beta }^{ac}{\omega }^b\wedge {\omega }_c^{\beta }+R_{\alpha \beta \gamma }^{ab}{\omega }^{\beta }\wedge {\omega }_b^{\gamma }+R_{\alpha \beta \gamma }^{abc}{\omega }_b^{\beta }\wedge {\omega }_c^{\gamma },\\ D{\tilde{\omega }}_{\alpha }={\tilde{\omega }}_{\alpha }^a\wedge {\tilde{\omega }}_a+{\tilde{\omega }}_{\alpha }^{\beta }\wedge {\tilde{\omega }}_{\beta }+R_{\alpha \beta \gamma }{\omega }^{\beta }\wedge {\omega }^{\gamma }+R_{\alpha ab}{\omega }^a\wedge {\omega }^b+R_{\alpha \beta a}{\omega }^{\beta }\wedge {\omega }^a+\\ K_{\alpha \beta \gamma }^a{\omega }^{\beta }\wedge {\omega }_a^{\gamma }+R_{\alpha a\beta }^b{\omega }^a\wedge {\omega }_b^{\beta }+K_{\alpha \beta \gamma }^{ab}{\omega }_a^{\beta }\wedge {\omega }_b^{\gamma }.\end{array}$$

The components of the curvature object have the form

$$\begin{array}{c}{R}_{b\alpha \beta }^a=L_{b\left[\alpha \beta \right]}^a-L_{b[\alpha }^c{L}_{c\beta ]}^a+{\Gamma }_{b[\alpha }^{ac}{g}_{c\beta ]},R_{bce}^a=L_{b\left[ce\right]}^a-L_{b[c}^d{L}_{de]}^a-{\delta }_{[c}^a{g}_{be]}-{\delta }_b^a{g}_{\left[ce\right]},\\ R_{b\alpha c}^a=L_{b\alpha c}^a-L_{bc\alpha }^a-L_{b\alpha }^e{L}_{ec}^a+L_{bc}^e{L}_{e\alpha }^a+{\Gamma }_{b\alpha }^{ae}{g}_{ec}+{\delta }_c^a{g}_{b\alpha }+{\delta }_b^a{g}_{c\alpha },\\ R_{b\alpha \beta }^{ac}=L_{b\alpha \beta }^{ac}-{\Gamma }_{b\beta \alpha }^{ac}-L_{b\alpha }^e{\Gamma }_{e\beta }^{ac}+{\Gamma }_{b\beta }^{ec}{L}_{e\alpha }^a+{\Gamma }_{b\alpha }^{ae}{g}_{e\beta }^c,\\ R_{bc\alpha }^{ad}=L_{bc\alpha }^{ad}-{\Gamma }_{b\alpha c}^{ad}-{\delta }_c^d{L}_{b\alpha }^a-L_{bc}^e{\Gamma }_{e\alpha }^{ad}+{\Gamma }_{b\alpha }^{ed}{L}_{ec}^a-{\delta }_c^a{g}_{b\alpha }^d-{\delta }_b^a{g}_{c\alpha }^d,R_{b\alpha \beta }^{acd}={\Gamma }_b^a{[}_{\alpha \beta }^{cd}]-{\Gamma }_b^e{[}_{\alpha }^c{\Gamma }_{e\beta }^{ad}],\\ R_{\beta \gamma \mu }^{\alpha }=L_{\beta \left[\gamma \mu \right]}^{\alpha }-L_{\beta [\gamma }^{\eta }{L}_{\eta \mu ]}^{\alpha }+{\Gamma }_{\beta [\gamma }^{\alpha a}{g}_{a\mu ]},R_{\beta ab}^{\alpha }=L_{\beta \left[ab\right]}^{\alpha }-L_{\beta [a}^{\gamma }{L}_{\gamma b]}^{\alpha }-{\delta }_{\beta }^{\alpha }{g}_{\left[ab\right]},\\ R_{\beta \gamma a}^{\alpha }=L_{\beta \gamma a}^{\alpha }-L_{\beta a\gamma }^{\alpha }-L_{\beta \gamma }^{\mu }{L}_{\mu a}^{\alpha }+L_{\beta a}^{\mu }{L}_{\mu \gamma }^{\alpha }+{\Gamma }_{\beta \gamma }^{\alpha b}{g}_{ba}+{\delta }_{\beta }^{\alpha }{g}_{a\gamma },\\ R_{\beta \gamma \mu }^{\alpha a}=L_{\beta \gamma \mu }^{\alpha a}-{\Gamma }_{\beta \mu \gamma }^{\alpha a}-L_{\beta \gamma }^{\eta }{\Gamma }_{\eta \mu }^{\alpha a}+{\Gamma }_{\beta \mu }^{\eta a}{L}_{\eta \gamma }^{\alpha }+{\Gamma }_{\beta \gamma }^{\alpha b}{g}_{b\mu }^a,\\ R_{\beta a\gamma }^{\alpha b}=L_{\beta a\gamma }^{\alpha b}-{\Gamma }_{\beta \gamma a}^{\alpha b}-L_{\beta a}^{\mu }{\Gamma }_{\mu \gamma }^{\alpha b}+{\Gamma }_{\beta \gamma }^{\mu b}{L}_{\mu a}^{\alpha }-{\delta }_a^b{L}_{\beta \gamma }^{\alpha }-{\delta }_{\beta }^{\alpha }{g}_{a\gamma }^b,R_{\beta \gamma \mu }^{\alpha ab}={\Gamma }_{\beta }^{\alpha }{[}_{\gamma \mu }^{ab}]-{\Gamma }_{\beta }^{\eta }{[}_{\gamma }^a{\Gamma }_{\eta \mu }^{\alpha b}],\\ R_{\alpha \beta \gamma }^a=L_{\alpha \left[\beta \gamma \right]}^a-L_{\alpha [\beta }^b{L}_{b\gamma ]}^a-L_{\alpha [\beta }^{\mu }{L}_{\mu \gamma ]}^a+{\Gamma }_{\alpha [\beta }^{ab}{g}_{b\gamma ]},R_{\alpha bc}^a=L_{\alpha \left[bc\right]}^a-L_{\alpha [b}^e{L}_{ec]}^a-L_{\alpha [b}^{\beta }{L}_{\beta c]}^a,\\ R_{\alpha \beta b}^a=L_{\alpha \beta b}^a-L_{\alpha b\beta }^a-L_{\alpha \beta }^e{L}_{eb}^a+L_{\alpha b}^c{L}_{c\beta }^a-L_{\alpha \beta }^{\gamma }{L}_{\gamma b}^a+L_{\alpha b}^{\mu }{L}_{\mu \beta }^a+{\Gamma }_{\alpha \beta }^{ac}{g}_{cb},\\ R_{\alpha b\beta }^{ac}=L_{\alpha b\beta }^{ac}-{\Gamma }_{\alpha \beta b}^{ac}-L_{\alpha b}^e{\Gamma }_{e\beta }^{ac}+{\Gamma }_{\alpha \beta }^{ec}{L}_{eb}^a-L_{\alpha b}^{\gamma }{\Gamma }_{\gamma \beta }^{ac}+{\Gamma }_{\alpha \beta }^{\gamma c}{L}_{\gamma b}^a-{\delta }_b^c{L}_{\alpha \beta }^a,\\ R_{\alpha \beta \gamma }^{ab}=L_{\alpha \beta \gamma }^{ab}-{\Gamma }_{\alpha \gamma \beta }^{ab}-L_{\alpha \beta }^c{\Gamma }_{c\gamma }^{ab}+{\Gamma }_{\alpha \gamma }^{cb}{L}_{c\beta }^a-L_{\alpha \beta }^{\mu }{\Gamma }_{\mu \gamma }^{ab}+{\Gamma }_{\alpha \gamma }^{\mu b}{L}_{\mu \beta }^a+{\Gamma }_{\alpha \beta }^{ac}{g}_{c\gamma }^b,\\ R_{\alpha \beta \gamma }^{abc}={\Gamma }_{\alpha }^a{[}_{\beta \gamma }^{bc}]-{\Gamma }_{\alpha }^e{[}_{\beta }^b{\Gamma }_{e\gamma }^{ac}]-{\Gamma }_{\alpha }^{\mu }{[}_{\beta }^b{\Gamma }_{\mu \gamma }^{ac}],R_{\alpha \beta \gamma }={\Gamma }_{\alpha \left[\beta \gamma \right]}-L_{\alpha [\beta }^{\mu }{\Gamma }_{\mu \gamma ]}+{\Pi }_{\alpha [\beta }^a{g}_{a\gamma ]},\\ R_{\alpha ab}={\Gamma }_{\alpha \left[ab\right]}-L_{\alpha [a}^{\beta }{\Gamma }_{\beta b]},R_{\alpha \beta a}={\Gamma }_{\alpha \beta a}-{\Gamma }_{\alpha a\beta }-L_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma a}+L_{\alpha a}^{\gamma }{\Gamma }_{\gamma \beta }+{\Pi }_{\alpha \beta }^b{g}_{ba},\\ K_{\alpha \beta \gamma }^a={\Gamma }_{\alpha \beta \gamma }^a-{\Pi }_{\alpha \gamma \beta }^a-L_{\alpha \beta }^{\mu }{\Pi }_{\mu \gamma }^a+{\Gamma }_{\alpha \gamma }^{\mu a}{\Gamma }_{\mu \beta }+{\Pi }_{\alpha \beta }^b{g}_{b\gamma }^a,\\ R_{\alpha a\beta }^b={\Gamma }_{\alpha a\beta }^b-{\Pi }_{\alpha \beta a}^b-L_{\alpha a}^{\gamma }{\Pi }_{\gamma \beta }^b+{\Gamma }_{\alpha \beta }^{\gamma b}{\Gamma }_{\gamma a}-{\delta }_a^b{\Gamma }_{\alpha \beta },K_{\alpha \beta \gamma }^{ab}={\Pi }_{\alpha }{[}_{\beta \gamma }^{ab}]-{\Gamma }_{\alpha }^{\mu }{[}_{\beta }^a{\Pi }_{\mu \gamma }^b].\end{array}$$

Theorem 8.

The curvature R is reduced to the object $R^2=\{R_{b\alpha \beta }^a$, $R_{bce}^a$, $R_{b\alpha c}^a$, $R_{b\alpha \beta }^{ac}$, $R_{bc\alpha }^{ad}$, $R_{b\alpha \beta }^{acd}$, $R_{\beta \gamma \mu }^{\alpha }$, $R_{\beta ab}^{\alpha }$, $R_{\beta \gamma a}^{\alpha }$, $R_{\beta \gamma \mu }^{\alpha a}$, $R_{\beta a\gamma }^{ab}$, $R_{\beta \gamma \mu }^{\alpha ab}$, $R_{\alpha \beta \gamma }^a$, $R_{\alpha bc}^a$, $R_{\alpha \beta b}^a$, $R_{\alpha b\beta }^{ac}$, $R_{\alpha \beta \gamma }^{ab}$, $R_{\alpha \beta \gamma }^{abc}$, $R_{\alpha \beta \gamma }$, $R_{\alpha ab}$, $R_{\alpha \beta a}$, $K_{\alpha \beta \gamma }^a$, $R_{\alpha a\beta }^b$, $K_{\alpha \beta \gamma }^{ab}\}$; and the reduced curvature object $R^2$ of the semi-normalized space ${\Pi }^2$ is a quasi-tensor together with the quasi-tensor $\{g_{a\alpha }$, $g_{ab}$, $g_{a\alpha }^b\}$. The object $R^2$ contains three subtensors that are curvature objects of subconnections on the bundles of plane and normal linear frames, and also on the bundle $H\left({\Pi }^2\right)$, where H is an affine quotent group of the stationary group of the pair $\{A,N_{m-1}\}$.

Proof of Theorem 8.

The proof of this theorem is not fundamentally different from the proof of Theorem 7, but now we use conditions (15). We obtain

$$\begin{array}{c}\hfill \Delta R_{b\alpha \beta }^a-R_{b[\alpha c}^a{\omega }_{\beta ]}^c\equiv 0,\Delta R_{bce}^a\equiv 0,\Delta R_{b\alpha c}^a+2R_{bce}^a{\omega }_{\alpha }^e\equiv 0,\Delta R_{b\alpha \beta }^{ac}-R_{be\beta }^{ac}{\omega }_{\alpha }^e\equiv 0,\\ \hfill \Delta R_{bc\alpha }^{ad}\equiv 0,\Delta R_{b\alpha \beta }^{ace}\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha }-R_{\beta [\gamma a}^{\alpha }{\omega }_{\mu ]}^a\equiv 0,\Delta R_{\beta ab}^{\alpha }\equiv 0,\\ \hfill \Delta R_{\beta \gamma a}^{\alpha }+2R_{\beta ab}^{\alpha }{\omega }_{\gamma }^b\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha a}-R_{\beta b\mu }^{\alpha a}{\omega }_{\gamma }^b\equiv 0,\Delta R_{\beta a\gamma }^{\alpha b}\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha ab}\equiv 0,\\ \hfill \Delta R_{\alpha \beta \gamma }^a-R_{\alpha [\beta b}^a{\omega }_{\gamma ]}^b-R_{b\beta \gamma }^a{\omega }_{\alpha }^b+R_{\alpha \beta \gamma }^{\mu }{\omega }_{\mu }^a\equiv 0,\Delta R_{\alpha bc}^a-R_{ebc}^a{\omega }_{\alpha }^e+R_{\alpha bc}^{\beta }{\omega }_{\beta }^a\equiv 0,\\ \hfill \Delta R_{\alpha \beta b}^a-2R_{\alpha cb}^a{\omega }_{\beta }^c-R_{c\beta b}^a{\omega }_{\alpha }^c+R_{\alpha \beta b}^{\gamma }{\omega }_{\gamma }^a\equiv 0,\Delta R_{\alpha b\beta }^{ac}-R_{eb\beta }^{ac}{\omega }_{\alpha }^e+R_{\alpha b\beta }^{\gamma c}{\omega }_{\gamma }^a\equiv 0,\\ \hfill \Delta R_{\alpha \beta \gamma }^{ab}-R_{\alpha c\gamma }^{ab}{\omega }_{\beta }^c+R_{\alpha \beta \gamma }^{\mu b}{\omega }_{\mu }^a-R_{c\beta \gamma }^{ab}{\omega }_{\alpha }^c\equiv 0,\Delta R_{\alpha \beta \gamma }^{abc}-R_{e\beta \gamma }^{abc}{\omega }_{\alpha }^e+R_{\alpha \beta \gamma }^{\mu bc}{\omega }_{\mu }^a\equiv 0,\\ \hfill \Delta R_{\alpha \beta \gamma }+R_{\alpha \beta \gamma }^{\mu }{\omega }_{\mu }-R_{\alpha [\beta a}{\omega }_{\gamma ]}^a-g_{a\left[\beta \gamma \right]}{\omega }_{\alpha }^a-g_{b[\beta }^a{g}_{a\gamma ]}{\omega }_{\alpha }^b\equiv 0,\Delta R_{\alpha ab}+R_{\alpha ab}^{\beta }{\omega }_{\beta }-g_{c\left[ab\right]}{\omega }_{\alpha }^c\equiv 0,\\ \hfill \Delta R_{\alpha \beta a}+2R_{\alpha ab}{\omega }_{\beta }^b+R_{\alpha \beta a}^{\gamma }{\omega }_{\gamma }+(g_{ba\beta }-g_{b\beta a}-g_{b\beta }^c{g}_{ca}){\omega }_{\alpha }^b\equiv 0,\\ \hfill \Delta K_{\alpha \beta \gamma }^a-R_{\alpha b\gamma }^a{\omega }_{\beta }^b+R_{\alpha \beta \gamma }^{\mu a}{\omega }_{\mu }+(g_{b\gamma \beta }^a-g_{b\beta \gamma }^a-g_{b\beta }^c{g}_{c\gamma }^a){\omega }_{\alpha }^b\equiv 0,\\ \hfill \Delta R_{\alpha a\beta }^b+R_{\alpha a\beta }^{\gamma b}{\omega }_{\gamma }+(g_{c\beta a}^b-g_{ca\beta }^b+{\delta }_a^b{g}_{c\beta }){\omega }_{\alpha }^c\equiv 0,\Delta K_{\alpha \beta \gamma }^{ab}+R_{\alpha \beta \gamma }^{\mu ab}{\omega }_{\mu }-g_c{[}_{\beta \gamma }^{ab}]{\omega }_{\alpha }^c\equiv 0.\end{array}$$

These differential congruences prove the theorem.  □

6.1.3. Curvature Object at the Full Canonization

At the normalization of the space $\Pi$, that is, at the conditions $A_{\alpha }\in N_{n-m}$ and $A_a\in N_{m-1}$, we have

$$\begin{array}{c}\hfill D{\tilde{\omega }}_b^a={\tilde{\omega }}_b^c\wedge {\tilde{\omega }}_c^a+R_{b\alpha \beta }^a{\omega }^{\alpha }\wedge {\omega }^{\beta }+R_{bce}^a{\omega }^c\wedge {\omega }^e+R_{b\alpha c}^a{\omega }^{\alpha }\wedge {\omega }^c+\\ \hfill R_{b\alpha \beta }^{ac}{\omega }^{\alpha }\wedge {\omega }_c^{\beta }+R_{bc\alpha }^{ad}{\omega }^c\wedge {\omega }_d^{\alpha }+R_{b\alpha \beta }^{acd}{\omega }_c^{\alpha }\wedge {\omega }_d^{\beta },\\ \hfill D{\tilde{\omega }}_{\beta }^{\alpha }={\tilde{\omega }}_{\beta }^{\gamma }\wedge {\tilde{\omega }}_{\gamma }^{\alpha }+R_{\beta \gamma \mu }^{\alpha }{\omega }^{\gamma }\wedge {\omega }^{\mu }+R_{\beta ab}^{\alpha }{\omega }^a\wedge {\omega }^b+R_{\beta \gamma a}^{\alpha }{\omega }^{\gamma }\wedge {\omega }^a+\\ \hfill R_{\beta \gamma \mu }^{\alpha a}{\omega }^{\gamma }\wedge {\omega }_a^{\mu }+R_{\beta a\gamma }^{\alpha b}{\omega }^a\wedge {\omega }_b^{\gamma }+R_{\beta \gamma \mu }^{\alpha ab}{\omega }_a^{\gamma }\wedge {\omega }_b^{\mu },\\ \hfill D{\tilde{\omega }}_{\alpha }={\tilde{\omega }}_{\alpha }^a\wedge {\tilde{\omega }}_a+{\tilde{\omega }}_{\alpha }^{\beta }\wedge {\tilde{\omega }}_{\beta }+R_{\alpha \beta \gamma }{\omega }^{\beta }\wedge {\omega }^{\gamma }+R_{\alpha ab}{\omega }^a\wedge {\omega }^b+R_{\alpha \beta a}{\omega }^{\beta }\wedge {\omega }^a+\\ \hfill K_{\alpha \beta \gamma }^a{\omega }^{\beta }\wedge {\omega }_a^{\gamma }+R_{\alpha a\beta }^b{\omega }^a\wedge {\omega }_b^{\beta }+K_{\alpha \beta \gamma }^{ab}{\omega }_a^{\beta }\wedge {\omega }_b^{\gamma }.\end{array}$$

The components of the curvature object are defined by the following relations:

$$\begin{array}{c}\hfill R_{b\alpha \beta }^a=L_{b\left[\alpha \beta \right]}^a-L_{b[\alpha }^c{L}_{c\beta ]}^a-L_{bc}^a{g}_{\alpha \beta ]}^c+{\Gamma }_{b[\alpha }^{ac}{g}_{c\beta ]},R_{bce}^a=L_{b\left[ce\right]}^a-L_{b[c}^d{L}_{de]}^a-{\delta }_{[c}^a{g}_{be]}-{\delta }_b^a{g}_{\left[ce\right]},\\ \hfill R_{b\alpha c}^a=L_{b\alpha c}^a-L_{bc\alpha }^a-L_{b\alpha }^e{L}_{ec}^a+L_{bc}^e{L}_{e\alpha }^a-L_{be}^a{g}_{\alpha c}^e+{\Gamma }_{b\alpha }^{ae}{g}_{ec}+{\delta }_c^a{g}_{b\alpha }+{\delta }_b^a{g}_{c\alpha },\\ \hfill R_{b\alpha \beta }^{ac}=L_{b\alpha \beta }^{ac}-{\Gamma }_{b\beta \alpha }^{ac}-L_{b\alpha }^e{\Gamma }_{e\beta }^{ac}+{\Gamma }_{b\beta }^{ec}{L}_{e\alpha }^a-L_{be}^a{g}_{\alpha \beta }^{ec}-{\delta }_b^c{g}_{\beta \alpha }^a+{\Gamma }_{b\alpha }^{ae}{g}_{e\beta }^c,\\ \hfill R_{bc\alpha }^{ad}=L_{bc\alpha }^{ad}-{\Gamma }_{b\alpha c}^{ad}-{\delta }_c^d{L}_{b\alpha }^a-L_{bc}^e{\Gamma }_{e\alpha }^{ad}+{\Gamma }_{b\alpha }^{ed}{L}_{ec}^a-{\delta }_b^d{g}_{\alpha c}^a-{\delta }_c^a{g}_{b\alpha }^d-{\delta }_b^a{g}_{c\alpha }^d,\\ \hfill R_{b\alpha \beta }^{acd}={\Gamma }_b^a{[}_{\alpha \beta }^{cd}]-{\Gamma }_b^e{[}_{\alpha }^c{\Gamma }_{e\beta }^{ad}]+{\delta }_b^{\lceil c}{g}_{\lfloor \alpha \beta }^{ad}],\\ \hfill R_{\beta \gamma \mu }^{\alpha }=L_{\beta \left[\gamma \mu \right]}^{\alpha }-L_{\beta [\gamma }^{\eta }{L}_{\eta \mu ]}^{\alpha }-L_{\beta a}^{\alpha }{g}_{\left[\gamma \mu \right]}^a+{\Gamma }_{\beta [\gamma }^{\alpha a}{g}_{a\mu ]},R_{\beta ab}^{\alpha }=L_{\beta \left[ab\right]}^{\alpha }-L_{\beta [a}^{\gamma }{L}_{\gamma b]}^{\alpha }-{\delta }_{\beta }^{\alpha }{g}_{\left[ab\right]},\\ \hfill R_{\beta \gamma a}^{\alpha }=L_{\beta \gamma a}^{\alpha }-L_{\beta a\gamma }^{\alpha }-L_{\beta \gamma }^{\mu }{L}_{\mu a}^{\alpha }+L_{\beta a}^{\mu }{L}_{\mu \gamma }^{\alpha }-L_{\beta b}^{\alpha }{g}_{\gamma a}^b+{\Gamma }_{\beta \gamma }^{\alpha b}{g}_{ba}+{\delta }_{\beta }^{\alpha }{g}_{a\gamma },\end{array}$$

$$\begin{array}{c}\hfill R_{\beta \gamma \mu }^{\alpha a}=L_{\beta \gamma \mu }^{\alpha a}-{\Gamma }_{\beta \mu \gamma }^{\alpha a}-L_{\beta \gamma }^{\eta }{\Gamma }_{\eta \mu }^{\alpha a}+{\Gamma }_{\beta \mu }^{\eta a}{L}_{\eta \gamma }^{\alpha }-L_{\beta b}^{\alpha }{g}_{\gamma \mu }^{ba}+{\delta }_{\mu }^{\alpha }{g}_{\beta \gamma }^a+{\Gamma }_{\beta \gamma }^{\alpha b}{g}_{b\mu }^a,\\ \hfill R_{\beta a\gamma }^{\alpha b}=L_{\beta a\gamma }^{\alpha b}-{\Gamma }_{\beta \gamma a}^{\alpha b}-L_{\beta a}^{\mu }{\Gamma }_{\mu \gamma }^{\alpha b}+{\Gamma }_{\beta \gamma }^{\mu b}{L}_{\mu a}^{\alpha }-{\delta }_a^b{L}_{\beta \gamma }^{\alpha }+{\delta }_{\gamma }^{\alpha }{g}_{\beta a}^b-{\delta }_{\beta }^{\alpha }{g}_{a\gamma }^b,\\ \hfill R_{\beta \gamma \mu }^{\alpha ab}={\Gamma }_{\beta }^{\alpha }{[}_{\gamma \mu }^{ab}]-{\Gamma }_{\beta }^{\eta }{[}_{\gamma }^a{\Gamma }_{\eta \mu }^{\alpha b}]-{\delta }_{\lfloor \gamma }^{\alpha }{g}_{\beta \mu }^{\lceil ab}],\\ \hfill R_{\alpha \beta \gamma }={\Gamma }_{\alpha \left[\beta \gamma \right]}-L_{\alpha [\beta }^{\mu }{\Gamma }_{\mu \gamma ]}-{\Gamma }_{\alpha a}{g}_{\left[\beta \gamma \right]}^a+g_{\alpha [\beta }^a{g}_{a\gamma ]}+{\Pi }_{\alpha [\beta }^a{g}_{a\gamma ]},R_{\alpha ab}={\Gamma }_{\alpha \left[ab\right]}-L_{\alpha [a}^{\beta }{\Gamma }_{\beta b]}+g_{\alpha [a}^c{g}_{cb]},\\ \hfill R_{\alpha \beta a}={\Gamma }_{\alpha \beta a}-{\Gamma }_{\alpha a\beta }-L_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma a}+L_{\alpha a}^{\gamma }{\Gamma }_{\gamma \beta }-{\Gamma }_{\alpha b}{g}_{\beta a}^b+g_{\alpha \beta }^b{g}_{ba}-g_{\alpha a}^b{g}_{b\beta }+{\Pi }_{\alpha \beta }^b{g}_{ba},\\ \hfill K_{\alpha \beta \gamma }^a={\Gamma }_{\alpha \beta \gamma }^a-{\Pi }_{\alpha \gamma \beta }^a-L_{\alpha \beta }^{\mu }{\Pi }_{\mu \gamma }^a+{\Gamma }_{\alpha \gamma }^{\mu a}{\Gamma }_{\mu \beta }-{\Gamma }_{\alpha b}{g}_{\beta \gamma }^{ba}+g_{\alpha \beta }^b{g}_{b\gamma }^a-g_{\alpha \gamma }^{ba}{g}_{b\beta }+{\Pi }_{\alpha \beta }^b{g}_{b\gamma }^a,\\ \hfill R_{\alpha a\beta }^b={\Gamma }_{\alpha a\beta }^b-{\Pi }_{\alpha \beta a}^b-L_{\alpha a}^{\gamma }{\Pi }_{\gamma \beta }^b+{\Gamma }_{\alpha \beta }^{\gamma b}{\Gamma }_{\gamma a}-{\delta }_a^b{\Gamma }_{\alpha \beta }+g_{\alpha a}^c{g}_{c\beta }^b-g_{\alpha \beta }^{cb}{g}_{ca},\\ \hfill K_{\alpha \beta \gamma }^{ab}={\Pi }_{\alpha }{[}_{\beta \gamma }^{ab}]-{\Gamma }_{\alpha }^{\mu }{[}_{\beta }^a{\Pi }_{\mu \gamma }^b]+g_{\alpha }^c{[}_{\beta }^a{g}_{c\gamma }^b].\end{array}$$

Theorem 9.

The curvature R is reduced to the object $R^{1,2}=\{R_{b\alpha \beta }^a$, $R_{bce}^a$, $R_{b\alpha c}^a$, $R_{b\alpha \beta }^{ac}$, $R_{bc\alpha }^{ad}$, $R_{b\alpha \beta }^{acd}$, $R_{\beta \gamma \mu }^{\alpha }$, $R_{\beta ab}^{\alpha }$, $R_{\beta \gamma a}^{\alpha }$, $R_{\beta \gamma \mu }^{\alpha a}$, $R_{\beta a\gamma }^{ab}$, $R_{\beta \gamma \mu }^{\alpha ab}$, $R_{\alpha \beta \gamma }$, $R_{\alpha ab}$, $R_{\alpha \beta a}$, $K_{\alpha \beta \gamma }^a$, $R_{\alpha a\beta }^b$, $K_{\alpha \beta \gamma }^{ab}\}$. The curvature object $R^{1,2}$ of the normalized space ${\Pi }^{1,2}$ is a tensor. The tensor $R^{1,2}$ contains two subtensors on the bundles of plane and normal linear frames, and also one curvature subtensor on the bundle of normal co-affine frames.

Proof of Theorem 9.

Indeed,

$$\begin{array}{c}\hfill \Delta R_{b\alpha \beta }^a\equiv 0,\Delta R_{bce}^a\equiv 0,\Delta R_{b\alpha c}^a\equiv 0,\Delta R_{b\alpha \beta }^{ac}\equiv 0,\Delta R_{bc\alpha }^{ad}\equiv 0,\Delta R_{b\alpha \beta }^{acd}\equiv 0,\\ \hfill \Delta R_{\beta \gamma \mu }^{\alpha }\equiv 0,\Delta R_{\beta ab}^{\alpha }\equiv 0,\Delta R_{\beta \gamma a}^{\alpha }\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha a}\equiv 0,\Delta R_{\beta a\gamma }^{\alpha b}\equiv 0,\Delta R_{\beta \gamma \mu }^{\alpha ab}\equiv 0,\\ \hfill \Delta R_{\alpha \beta \gamma }+R_{\alpha \beta \gamma }^{\mu }{\omega }_{\mu }\equiv 0,\Delta R_{\alpha ab}+R_{\alpha ab}^{\beta }{\omega }_{\beta }\equiv 0,\Delta R_{\alpha \beta a}+R_{\alpha \beta a}^{\gamma }{\omega }_{\gamma }\equiv 0,\\ \hfill \Delta K_{\alpha \beta \gamma }^a+R_{\alpha \beta \gamma }^{\mu a}{\omega }_{\mu }\equiv 0,\Delta R_{\alpha a\beta }^b+R_{\alpha a\beta }^{\gamma b}{\omega }_{\gamma }\equiv 0,\Delta K_{\alpha \beta \gamma }^{ab}+R_{\alpha \beta \gamma }^{\mu ab}{\omega }_{\mu }\equiv 0.\end{array}$$

These congruences conclude the proof.  □

6.2. Torsion Objects

The following equations are a result of substituting the connection forms (23) into the structure equations of basis forms of the space $\Pi$.

$$\begin{array}{c}\hfill D{\omega }^{\alpha }={\omega }^a\wedge {\omega }_a^{\alpha }-{\tilde{\omega }}_{\beta }^{\alpha }\wedge {\omega }^{\beta }+S_{\beta \gamma }^{\alpha }{\omega }^{\beta }\wedge {\omega }^{\gamma }+S_{\beta a}^{\alpha }{\omega }^{\beta }\wedge {\omega }^a+S_{\beta \gamma }^{\alpha a}{\omega }^{\beta }\wedge {\omega }_a^{\gamma },\\ \hfill D{\omega }^a={\omega }^b\wedge {\tilde{\omega }}_b^a+{\omega }^{\alpha }\wedge {\tilde{\omega }}_{\alpha }^a+S_{bc}^a{\omega }^b\wedge {\omega }^c+S_{b\alpha }^a{\omega }^b\wedge {\omega }^{\alpha }+S_{\alpha \beta }^a{\omega }^{\alpha }\wedge {\omega }^{\beta }+S_{b\alpha }^{ac}{\omega }^b\wedge {\omega }_c^{\alpha }+S_{\alpha \beta }^{ab}{\omega }^{\alpha }\wedge {\omega }_b^{\beta },\\ \hfill D{\omega }_a^{\alpha }={\tilde{\omega }}_a^b\wedge {\omega }_b^{\alpha }-{\tilde{\omega }}_{\beta }^{\alpha }\wedge {\omega }_a^{\beta }+{\tilde{\omega }}_a\wedge {\omega }^{\alpha }+S_{a\beta \gamma }^{\alpha }{\omega }^{\beta }\wedge {\omega }^{\gamma }+S_{ab\beta }^{\alpha }{\omega }^b\wedge {\omega }^{\beta }+S_{a\beta \gamma }^{\alpha b}{\omega }^{\beta }\wedge {\omega }_b^{\gamma }+\\ \hfill S_{a\beta c}^{\alpha b}{\omega }^c\wedge {\omega }_b^{\beta }+S_{a\beta \gamma }^{\alpha bc}{\omega }_b^{\beta }\wedge {\omega }_c^{\gamma },\end{array}$$

where the object’s S components are defined as follows:

$$\begin{array}{c}\hfill S_{\beta \gamma }^{\alpha }=L_{\left[\beta \gamma \right]}^{\alpha },S_{\beta a}^{\alpha }=L_{\beta a}^{\alpha },S_{\beta \gamma }^{\alpha a}={\Gamma }_{\beta \gamma }^{\alpha a},S_{bc}^a=L_{\left[bc\right]}^a,S_{b\alpha }^a=L_{b\alpha }^a-L_{\alpha b}^a,S_{\alpha \beta }^a=L_{\left[\alpha \beta \right]}^a,\\ \hfill S_{b\alpha }^{ac}={\Gamma }_{b\alpha }^{ac},S_{\alpha \beta }^{ab}={\Gamma }_{\alpha \beta }^{ab},S_{a\beta \gamma }^{\alpha }={\delta }_{[\gamma }^{\alpha }{\Gamma }_{a\beta ]},S_{ab\gamma }^{\alpha }={\delta }_{\gamma }^{\alpha }{\Gamma }_{ab},\\ \hfill S_{a\beta \gamma }^{\alpha b}={\delta }_{\gamma }^{\alpha }{L}_{a\beta }^b-{\delta }_a^b{L}_{\gamma \beta }^{\alpha }-{\delta }_{\beta }^{\alpha }{\Pi }_{a\gamma }^b,S_{a\beta c}^{\alpha b}={\delta }_{\beta }^{\alpha }{L}_{ac}^b-{\delta }_a^b{L}_{\beta c}^{\alpha },S_{a\beta \gamma }^{\alpha bc}=-{\delta }_{\lfloor \beta }^{\alpha }{\Gamma }_{a\gamma }^{\lceil bc}]+{\delta }_a^{\lceil b}{\Gamma }_{\lfloor \beta \gamma }^{\alpha c}].\end{array}$$

(30)

The right-hand sides of equalities (30) contain only components of the subobject ${\Gamma }_1$, and, therefore, let object S be a torsion object of linear subconnection ${\Gamma }_1$ of the group connection $\Gamma$ in the space $\Pi$ of centered planes $P_m^0$.

Taking into account congruences (24) for the components of subobject ${\Gamma }_1$, the congruences modulo the basis forms

$$\begin{array}{c}\hfill \Delta S_{\beta \gamma }^{\alpha }+S_{\left[\beta \gamma \right]}^{\alpha a}{\omega }_a-S_{[\beta a}^{\alpha }{\omega }_{\gamma ]}^a\equiv 0,\Delta S_{\beta a}^{\alpha }-{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta S_{\beta \gamma }^{\alpha a}-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }^a\equiv 0,\Delta S_{bc}^a\equiv 0,\\ \hfill \Delta S_{b\alpha }^a-2S_{bc}^a{\omega }_{\alpha }^c+S_{b\alpha }^{ac}{\omega }_c-S_{\alpha b}^{\beta }{\omega }_{\beta }^a\equiv 0,\Delta S_{\alpha \beta }^a+S_{b[\alpha }^a{\omega }_{\beta ]}^b+S_{\alpha \beta }^{\gamma }{\omega }_{\gamma }^a+S_{\left[\alpha \beta \right]}^{ab}{\omega }_b\equiv 0,\\ \hfill \Delta S_{b\alpha }^{ac}+{\delta }_b^c{\omega }_{\alpha }^a\equiv 0,\Delta S_{\alpha \beta }^{ab}-S_{c\beta }^{ab}{\omega }_{\alpha }^c+S_{\alpha \beta }^{\gamma b}{\omega }_{\gamma }^a\equiv 0,\Delta S_{a\beta \gamma }^{\alpha }+S_{ab[\beta }^{\alpha }{\omega }_{\gamma ]}^b+S_{a\left[\beta \gamma \right]}^{\alpha b}{\omega }_b-S_{\beta \gamma }^{\alpha }{\omega }_a\equiv 0,\\ \hfill \Delta S_{ab\beta }^{\alpha }+S_{a\beta b}^{\alpha c}{\omega }_c+S_{\beta b}^{\alpha }{\omega }_a\equiv 0,\Delta S_{a\beta c}^{\alpha b}-{\delta }_c^b{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta S_{a\beta \gamma }^{\alpha bc}\equiv 0,\\ \hfill \Delta S_{a\beta \gamma }^{\alpha b}-S_{a\gamma c}^{\alpha b}{\omega }_{\beta }^c-2{\delta }_{\lfloor \beta }^{\alpha }{S}_{a\gamma }^{\lceil cb}]{\omega }_c-{\delta }_a^b{S}_{\gamma \beta }^{\alpha c}{\omega }_c\equiv 0\end{array}$$

(31)

are obtained. Now, it is evident that the following theorem holds.

Theorem 10.

The torsion object S of the subconnection ${\Gamma }_1$ is a quasi-tensor containing the tensors $S_{bc}^a$, $S_{a\beta \gamma }^{\alpha bc}$ and the quasi-tensors $S_{\beta a}^{\alpha }$, $S_{\beta \gamma }^{\alpha a}$, $S_{b\alpha }^{ac}$, $S_{b\beta c}^{\alpha a}$, $\{S_{\beta a}^{\alpha }$, $S_{\beta \gamma }^{\alpha a}$, $S_{\beta \gamma }^{\alpha }\}$, $\{S_{bc}^a$, $S_{b\alpha }^{ac}$, $S_{\alpha b}^{\beta }$, $S_{b\alpha }^a\}$, $\{S_{c\beta }^{ab}$, $S_{\alpha \beta }^{\gamma b}$, $S_{\alpha \beta }^{ab}\}$, $\{S_{a\beta b}^{\alpha c}$, $S_{\beta b}^{\alpha }$, $S_{ab\beta }^{\alpha }\}$, $\{S_{a\gamma c}^{\alpha b}$, $S_{a\gamma }^{cb}$, $S_{\gamma \beta }^{\alpha c}$, $S_{a\beta \gamma }^{\alpha b}\}$.

Remark 7.

Because of Theorem 10, the connection ${\Gamma }_1$ is always with torsion (see [32], cf. [33]) as the torsion object S of connection ${\Gamma }_1$ is a quasi-tensor.

It remains to consider the dynamics of changes of the torsion quasi-tensor S at consecutive canonizations.

6.2.1. The First Canonization

We put the vertices $A_{\alpha }$ on the first normal $N_{n-m}$; then, conditions (8) can be written as

$${\omega }_{\alpha }^a\equiv 0(mod{\omega }^{\alpha },{\omega }_a^{\alpha },{\omega }^a).$$

(32)

Using congruences (32), the differential congruences (31) will be written in the form:

$$\begin{array}{c}\hfill \Delta S_{\beta \gamma }^{\alpha }+S_{\left[\beta \gamma \right]}^{\alpha a}{\omega }_a\equiv 0,\Delta S_{\beta a}^{\alpha }-{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta S_{\beta \gamma }^{\alpha a}\equiv 0,\Delta S_{bc}^a\equiv 0,\\ \hfill \Delta S_{b\alpha }^a+S_{b\alpha }^{ac}{\omega }_c\equiv 0,\Delta S_{\alpha \beta }^a+S_{\left[\alpha \beta \right]}^{ab}{\omega }_b\equiv 0,\Delta S_{b\alpha }^{ac}\equiv 0,\Delta S_{\alpha \beta }^{ab}\equiv 0,\\ \hfill \Delta S_{a\beta \gamma }^{\alpha }+S_{a\left[\beta \gamma \right]}^{\alpha b}{\omega }_b-S_{\beta \gamma }^{\alpha }{\omega }_a\equiv 0,\Delta S_{ab\beta }^{\alpha }+S_{a\beta b}^{\alpha c}{\omega }_c+S_{\beta b}^{\alpha }{\omega }_a\equiv 0,\\ \hfill \Delta S_{a\beta c}^{\alpha b}-{\delta }_c^b{\delta }_{\beta }^{\alpha }{\omega }_a\equiv 0,\Delta S_{a\beta \gamma }^{\alpha bc}\equiv 0,\Delta S_{a\beta \gamma }^{\alpha b}-2{\delta }_{\lfloor \beta }^{\alpha }{S}_{a\gamma }^{\lceil cb}]{\omega }_c-{\delta }_a^b{S}_{\gamma \beta }^{\alpha c}{\omega }_c\equiv 0.\end{array}$$

(33)

Theorem 11.

Suppose that the components of the torsion quasi-tensor S satisfy congruences (31); then, S is reduced to the quasi-tensor $S^1$ with congruences (33) at the adaptation of the moving frame to a field of the first normals $N_{n-m}$. The quasi-tensor $S^1$ contains the tensors $S_{\beta \gamma }^{\alpha a}$, $S_{bc}^a$, $S_{b\alpha }^{ac}$, $S_{\alpha \beta }^{ab}$, $S_{a\beta \gamma }^{\alpha bc}$, $\{S_{\beta \gamma }^{\alpha a}$, $S_{\beta \gamma }^{\alpha }\}$, $\{S_{\alpha \beta }^{ab}$, $S_{\alpha \beta }^a\}$, $\{S_{b\alpha }^{ac}$, $S_{b\alpha }^a\}$, $\{S_{a\gamma }^{cb}$, $S_{\gamma \beta }^{\alpha c}$, $S_{a\beta \gamma }^{\alpha b}\}$ and the quasi-tensors $S_{\beta a}^{\alpha }$, $S_{a\beta c}^{\alpha b}$, $\{S_{a\gamma b}^{\alpha c}$, $S_{\gamma b}^{\alpha }$, $S_{ab\gamma }^{\alpha }\}$.

6.2.2. The Second Canonization

Without using the previous canonization, we put the vertices $A_a$ on the second normal $N_{m-1}$, then conditions (8) can be written as

$${\omega }_a\equiv 0(mod{\omega }^{\alpha },{\omega }_a^{\alpha },{\omega }^a).$$

(34)

Taking into account condition (34), the differential congruences for the components of the torsion object S have the form

$$\begin{array}{c}\hfill \Delta S_{\beta \gamma }^{\alpha }-S_{[\beta a}^{\alpha }{\omega }_{\gamma ]}^a\equiv 0,\Delta S_{\beta a}^{\alpha }\equiv 0,\Delta S_{\beta \gamma }^{\alpha a}-{\delta }_{\gamma }^{\alpha }{\omega }_{\beta }^a\equiv 0,\Delta S_{bc}^a\equiv 0,\\ \hfill \Delta S_{b\alpha }^a-2S_{bc}^a{\omega }_{\alpha }^c-S_{\alpha b}^{\beta }{\omega }_{\beta }^a\equiv 0,\Delta S_{\alpha \beta }^a+S_{b[\alpha }^a{\omega }_{\beta ]}^b+S_{\alpha \beta }^{\gamma }{\omega }_{\gamma }^a\equiv 0,\\ \hfill \Delta S_{b\alpha }^{ac}+{\delta }_b^c{\omega }_{\alpha }^a\equiv 0,\Delta S_{\alpha \beta }^{ab}-S_{c\beta }^{ab}{\omega }_{\alpha }^c+S_{\alpha \beta }^{\gamma b}{\omega }_{\gamma }^a\equiv 0,\Delta S_{a\beta \gamma }^{\alpha }+S_{ab[\beta }^{\alpha }{\omega }_{\gamma ]}^b\equiv 0,\\ \hfill \Delta S_{ab\beta }^{\alpha }\equiv 0,\Delta S_{a\beta \gamma }^{\alpha b}-S_{a\gamma c}^{\alpha b}{\omega }_{\beta }^c\equiv 0,\Delta S_{a\beta c}^{\alpha b}\equiv 0,\Delta S_{a\beta \gamma }^{\alpha bc}\equiv 0.\end{array}$$

(35)

Thus, Theorem 12 follows from these congruences.

Theorem 12.

Suppose that the components of the torsion quasi-tensor S satisfy congruences (31); then S is reduced to the quasi-tensor $S^2$ with congruences (35) at the adaptation of the moving frame to a field of the second normals $N_{m-1}$. The quasi-tensor $S^2$ contains the tensors $S_{\beta a}^{\alpha }$, $S_{bc}^a$, $S_{a\beta \gamma }^{\alpha bc}$, $S_{a\beta c}^{\alpha b}$, $S_{ab\gamma }^{\alpha }$, $\{S_{\beta a}^{\alpha }$, $S_{\beta \gamma }^{\alpha }\}$, $\{S_{bc}^a$, $S_{\alpha b}^{\beta }$, $S_{b\alpha }^a\}$, $\{S_{a\gamma c}^{\alpha b}$, $S_{a\beta \gamma }^{\alpha b}\}$, $\{S_{ab\beta }^{\alpha }$, $S_{a\beta \gamma }^{\alpha }\}$ and the quasi-tensors $S_{\beta \gamma }^{\alpha a}$, $S_{b\alpha }^{ac}$, $\{S_{c\beta }^{ab}$, $S_{\alpha \beta }^{\gamma b}$, $S_{\alpha \beta }^{ab}\}$.

6.2.3. The Full Canonization

After all, if $A_{\alpha }\in P_{n-m}$ and $A_a\in P_{m-1}$, then conditions (32) and (34) hold and the differential congruences of the torsion object components have the form

$$\begin{array}{c}\hfill \Delta S_{\beta \gamma }^{\alpha }\equiv 0,\Delta S_{\beta a}^{\alpha }\equiv 0,\Delta S_{\beta \gamma }^{\alpha a}\equiv 0,\Delta S_{bc}^a\equiv 0,\Delta S_{b\alpha }^a\equiv 0,\Delta S_{\alpha \beta }^a\equiv 0,\Delta S_{b\alpha }^{ac}\equiv 0,\\ \hfill \Delta S_{\alpha \beta }^{ab}\equiv 0,\Delta S_{a\beta \gamma }^{\alpha }\equiv 0,\Delta S_{ab\beta }^{\alpha }\equiv 0,\Delta S_{a\beta \gamma }^{\alpha b}\equiv 0,\Delta S_{a\beta c}^{\alpha b}\equiv 0,\Delta S_{a\beta \gamma }^{\alpha bc}\equiv 0.\end{array}$$

Theorem 13.

At the adaptation of the moving frame to the normalization of the space Π, the torsion quasi-tensor S becomes a tensor; furthermore, all its components are the one-component tensors (cf. in Theorems 11 and 12, the quasi-tensors $S^1$ and $S^2$ contain also multi-component tensors and quasi-tensors).

Remark 8.

It is true that, at the considered canonizations, the number of components of the torsion object S does not change, in contrast to the reduced objects of curvature and connection, yet the fact remains that differential congruences of its components differ from congruences (31). This testifies to possible change of properties of torsion for semi-normalized spaces ${\Pi }^1$, ${\Pi }^2$ and also for the normalized space ${\Pi }^{1,2}$.