Geometry of Sub-Riemannian Manifolds Equipped with a Quasi-Semi-Weyl Structure

Abstract

Quasi-semi-Weyl and quasi-statistical structures are based on a connection with torsion. In this chapter, as a connection with torsion, we consider the so-called extended connection, which is defined with the help of an intrinsic connection, i.e., a connection in the distribution of a sub-Riemannian manifold, as well as with the help of an endomorphism that preserves the indicated distribution and is called a structural endomorphism. It is proved that the extended connection is a connection of the quasi-semi-Weyl structure of a sub-Riemannian manifold of contact type only if the distribution of the sub-Riemannian manifold is involutive. In order to be able to consider sub-Riemannian manifolds with a not necessarily involutive distribution, the concepts of sub-Riemannian quasi-semi-Weyl and sub-Riemannian quasi-statistical structures are introduced, which are modifications of quasi-semi-Weyl and quasi-statistical structures for the case of sub-Riemannian manifolds of contact type. The structural endomorphism for the connection of a sub-Riemannian quasi-statistical structure is found. As an example, we consider non-holonomic Kenmotsu manifolds, which are sub-Riemannian manifolds of contact type endowed with an additional structure. It is proved that the restriction of the structural endomorphism to the distribution of such manifold differs from the identity transformation only by a factor.

1. Introduction

A torsion-free linear connection $\nabla$ on a manifold $M$ is said to be metric compatible with $g$ if the metric tensor is a Codacci tensor with respect to $\nabla$:

$${\nabla }_{X}g\left(Y,Z\right)={\nabla }_{Y}g\left(X,Z\right),X,Y,Z\in \mathrm{\Gamma }\left(\mathit{TM}\right).$$E1

The statistical structure introduced by Lauritzen in [1] is a pair $\left(g,\nabla \right)$, where $g$ is a pseudo-Riemannian metric and $\nabla$ is a torsion-free linear connection compatible with it. A manifold $M$ equipped with a statistical structure is called a statistical manifold. Statistical structures are related to the theory of conjugate linear connections developed by A.P. Norden [2]. The set of torsion-free connections conjugate with respect to the metric constitutes a very interesting class of connections compatible with the metric. This class, along with the Levi-Civita connection, which is the only self-adjoint connection, includes other connections that are of interest to researchers. Linear connections compatible with the Riemannian metric find interesting applications in the geometric interpretation of a number of problems in mathematical statistics [3, 4, 5].

To describe geometric structures in the spaces of quantum states, it is convenient to use quasi-statistical manifolds, assuming the presence of torsion in a connection compatible with the metric. In [6], the concept of a quasi-semi-Weyl structure was introduced and methods for constructing quasi-statistical and quasi-semi-Weyl structures were proposed.

In this chapter, analogues of the quasi-statistical and quasi-semi-Weyl structures are introduced, these are the sub-Riemannian quasi-statistical structure and the sub-Riemannian quasi-semi-Weyl structure, respectively. The refinement of previously known concepts is caused by the desire to take into account the specifics of sub-Riemannian manifolds of contact type, on which these structures are specified. A sub-Riemannian manifold of contact type is a smooth manifold $M$ of dimension $n$ with a sub-Riemannian structure $\left(M,\overrightarrow{\xi },\eta ,g,D\right)$ given on it, where $\eta$ is a 1-form generating the distribution $D$, $D=ker\left(\eta \right)$, $\overrightarrow{\xi }$ is a vector field that generates a rigging $D^{\perp }$ of the distribution $D$, $D^{\perp }=\mathit{\text{span}}\left(\overrightarrow{\xi }\right)$, $g$ is a Riemannian metric on the manifold $M$ with respect to which the distributions $D$ and $D^{\perp }$ are mutually orthogonal. In this case, the equalities $\eta \left(\overrightarrow{\xi }\right)=1$, $g\left(\overrightarrow{\xi },\overrightarrow{\xi }\right)=1$, $\mathit{d\eta }\left(\overrightarrow{\xi },\cdot \right)=0$ hold true.

A sub-Riemannian quasi-statistical manifold is a sub-Riemannian manifold of contact type endowed with a sub-Riemannian quasi-statistical structure $\left(M,g,{\nabla }^N\right)$. Here ${\nabla }^N$ is a connection with torsion of a special type, which is called in this work an extended connection, or an N-connection. An N-connection ${\nabla }^N$ is defined by a pair $\left(\nabla ,N\right)$, where $\nabla$ is an intrinsic linear connection [7], $N:\mathit{TM}\to \mathit{TM}$ is an endomorphism of the tangent bundle of the manifold $M$ such that $N\overrightarrow{\xi }=\overrightarrow{0}$, $N\left(D\right)\subset D$. We will say that the N-connection ${\nabla }^N$ is the extension of a connection $\nabla$ by means of the endomorphism $N$.

An extended connection ${\nabla }^N$ is defined as the only connection that satisfies the following conditions:

(1) ${\nabla }_X^{N}Y\in \mathrm{\Gamma }\left(\mathrm{D}\right)$, (2) ${\nabla }_X^N\overrightarrow{\xi }=\overrightarrow{0}$, (3) ${\nabla }_{\overrightarrow{\xi }}^{N}Y=\left[\overrightarrow{\xi },Y\right]+\mathit{NY}$, (4) ${\nabla }_Y^{N}Z={\nabla }_{Y}Z$, $X\in \mathrm{\Gamma }\left(\mathit{TM}\right);Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right)$. If ${\nabla }_{X}g\left(Y,Z\right)=0$, ${\nabla }_{X}Y-{\nabla }_{Y}X-P\left[X,Y\right]=0$, where $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right)$ and $P:D\oplus D^{\perp }\to D$ is a projector, then $\nabla$ is called an intrinsic metric connection. The torsion $T\left(X,Y\right)$ of an intrinsic linear connection is given by the equality $T\left(X,Y\right)={\nabla }_{X}Y-{\nabla }_{Y}X-P\left[X,Y\right]$.

It can be directly verified that the torsion $S\left(X,Y\right)$ of the connection ${\nabla }^N$ is given by the following formula [7, 8]:

$$S\left(X,Y\right)=2\omega \left(X,Y\right)\overrightarrow{\xi }+\eta \left(X\right)\mathit{NY}-\eta \left(Y\right)\mathit{NX},X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right).$$E2

Here $\omega \left(X,Y\right)=\mathit{d\eta }\left(X,Y\right).$

The interest to the connections with torsion is due to their use in theoretical physics [9, 10, 11]. An N-connection ${\nabla }^N$ was defined by one of the authors of this chapter and was studied by him, for example, in [7, 12, 13, 14, 15]. By appropriately specifying the endomorphism $N$, one can obtain most of the previously known classes of connections with torsion, e.g., the Schouten-van Kampen connection, the Tanaka-Webster connection, etc. [7, 16]. At the same time, in those papers in which connections with torsion ${\nabla }^N$ were used (for specific endomorphisms $N$), the presence of an endomorphism $N$ was not explicitly discussed. An exception is the work [17] (see also [13]), where the properties of the endomorphism $N$, denoted in the paper by the symbol “$\tau$”, received a special attention.

Motivation for defining and studying sub-Riemannian quasi-statistical structures $\left(M,g,{\nabla }^N\right)$ is supported by the following facts:

1. As mentioned above, the class of extended connections ${\nabla }^N$ is widely represented in modern geometric studies;

2. N-connections ${\nabla }^N$ arise naturally as continuations of the intrinsic connections $\nabla$, which occupy an important place in the geometric modeling of problems in non-holonomic mechanics and theoretical physics [7].

In the present work, we define a sub-Riemannian quasi-statistical structure as a triple $\left(M,g,{\nabla }^N\right)$, where $M$ is a sub-Riemannian manifold, and the connection ${\nabla }^N$ is related to the metric $g$ through the equality

$${\nabla }_X^{N}g\left(Y,Z\right)-{\nabla }_Y^{N}g\left(X,Z\right)+\tilde{S}\left(X,Y,Z\right)-2\omega \left(X,Y\right)\eta \left(Z\right)=0,$$E3

where $\tilde{S}\left(X,Y,Z\right)=g\left(S\left(X,Y\right),Z\right)$, $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right)$, $\omega =\mathit{d\eta }$.

The extension of the term “quasi-statistical structure” by adding the word “sub-Riemannian” is justified by the use in the structure-defining equality of the external form $\omega =\mathit{d\eta }$, referring to the intrinsic geometry of a sub-Riemannian manifold of contact type [7]. Although a sub-Riemannian quasi-statistical structure is defined on a sub-Riemannian manifold of contact type, as a basis manifold we will often use almost contact metric manifolds as sub-Riemannian manifolds carrying an additional structure.

2. Main results

An almost contact metric manifold is a smooth manifold $M$ of odd dimension $n=2m+\mathit{1}$, $m\ge 1$, with an almost contact metric structure $\left(M,\overrightarrow{\xi },\eta ,\phi ,g\right)$ defined on it [8, 16]. Here, in particular, $\eta$ is a 1-form and $\overrightarrow{\xi }$ is a vector field generating, respectively, the distribution $D$: $D=ker\left(\eta \right)$ and the rigging $D^{\perp }$ of the distribution $D$: $D^{\perp }=\mathit{\text{span}}\left(\overrightarrow{\xi }\right)$. A smooth distribution D is called the distribution of an almost contact metric manifold. There is the decomposition $\mathit{TM}=D\oplus D^{\perp }$. An almost contact metric manifold is called normal if the condition $N_{\phi }+2\mathit{d\eta }\otimes \overrightarrow{\xi }=0$ is satisfied, where $N_{\phi }\left(X,Y\right)=\left[\mathit{\phi X},\mathit{\phi Y}\right]+{\phi }^2\left[X,Y\right]-\phi \left[\mathit{\phi X},Y\right]-\phi \left[X,\mathit{\phi Y}\right]$ is the Nijenhuis tensor of the endomorphism $\mathrm{\phi }$.

To carry out the necessary calculations, it is convenient to use the adapted coordinates [8]. A chart $k\left(x^i\right)\left(i,j,k=1,\dots ,n;a,b,c=1,\dots ,n-1\right)$ on the manifold $M$ is said to be adapted to the distribution D if ${\partial }_{\mathrm{n}}=\overrightarrow{\xi }$ [7]. Let $P:\mathit{TM}\to D$ be the projection defined by the decomposition $\mathit{TM}=D\oplus D^{\perp }$ and $k\left(x^{\alpha }\right)$ be an adapted chart. The vector fields $P\left({\partial }_a\right)={\overrightarrow{e}}_a={\partial }_a-{\mathrm{\Gamma }}_a^n{\partial }_n$ generate the distribution $D:D=\mathit{\text{Span}}\left({\overrightarrow{e}}_a\right)$. For a non-holonomic frame field $\left({\overrightarrow{e}}_i\right)=\left({\overrightarrow{e}}_a,{\partial }_n\right)$, the relation $\left[{\overrightarrow{\mathrm{e}}}_{\mathrm{a}},{\overrightarrow{e}}_b\right]=2{\omega }_{\mathit{ba}}{\partial }_n$ holds true. We additionally require the condition $\omega \left(\overrightarrow{\xi },\cdot \right)=0$, which is equivalent to the condition ${\partial }_n{\mathrm{\Gamma }}_a^n=0$. The condition $\omega \left(\overrightarrow{\xi },\cdot \right)=0$ does not impose strong restrictions on the structures under consideration. For example, all normal almost contact metric manifolds satisfy this condition.

Let $k\left(x^i\right)$ and $k^{\prime }\left(x^{i^{\prime }}\right)$ be adapted charts, then we obtain the following coordinate transformation formulas:

$$x^a=x^a\left(x^{a^{\prime }}\right),x^n=x^{n^{\prime }}+x^n\left(x^{a^{\prime }}\right).$$E4

An intrinsic linear connection $\nabla$ [8] on an almost contact metric manifold is a map

$$\nabla :\mathrm{\Gamma }\left(D\right)\times \mathrm{\Gamma }\left(D\right)\to \mathrm{\Gamma }\left(D\right)$$E5

satisfying the following conditions:

1. ${\nabla }_{f_{1}X+f_{2}Y}=f_1{\nabla }_X+f_2{\nabla }_Y,$

2. ${\nabla }_X\mathit{fY}=\left(\mathit{Xf}\right)Y+f{\nabla }_{X}Y,$

3. ${\nabla }_X\left(Y+Z\right)={\nabla }_{X}Y+{\nabla }_{X}Z,$

where $\mathrm{\Gamma }\left(\mathrm{D}\right)$ is the module of admissible vector fields (vector fields belonging to the distribution $D$ at each point of the manifold).

Let $\tilde{\nabla }$ be the Levi-Civita connection and ${\tilde{\mathrm{\Gamma }}}_{\mathit{jk}}^i$ be its Christoffel symbols. Using equality

$$2{\mathrm{\Gamma }}_{\mathit{ij}}^m=g^{\mathit{km}}\left(e_i{g}_{\mathit{jk}}+e_j{g}_{\mathit{ik}}-e_k{g}_{\mathit{ij}}+{\mathrm{\Omega }}_{\mathit{kj}}^l{g}_{\mathit{li}}+{\mathrm{\Omega }}_{\mathit{ki}}^l{g}_{\mathit{lj}}\right)+{\mathrm{\Omega }}_{\mathit{ij}}^m,$$E6

where $\left[{\overrightarrow{\mathrm{e}}}_{\mathrm{i}},{\overrightarrow{e}}_j\right]={\mathrm{\Omega }}_{\mathit{ij}}^m{\overrightarrow{e}}_k$, we prove the following proposition [8].

Proposition 1. The Christoffel symbols ${\tilde{\mathrm{\Gamma }}}_{\mathit{ij}}^k$ of the Levi-Civita connection of a sub-Riemannian manifold with respect to the adapted coordinates have the following form:

$$\begin{array}{l}{\tilde{\mathrm{\Gamma }}}_{\mathit{ab}}^c={\mathrm{\Gamma }}_{\mathit{ab}}^c,{\tilde{\mathrm{\Gamma }}}_{\mathit{ab}}^n={\omega }_{\mathit{ba}}-C_{\mathit{ab}},{\tilde{\mathrm{\Gamma }}}_{\mathit{\text{an}}}^b={\tilde{\mathrm{\Gamma }}}_{\mathit{na}}^b=C_a^b+{\psi }_a^b,{\tilde{\mathrm{\Gamma }}}_{\mathit{na}}^n=-{\partial }_n{\mathrm{\Gamma }}_a^n,{\tilde{\mathrm{\Gamma }}}_{\mathit{nn}}^a=g^{\mathit{ab}}{\partial }_n{\mathrm{\Gamma }}_b^n,\\ \text{where}{\mathrm{\Gamma }}_{\mathit{bc}}^a=\frac{1}{2}{g}^{\mathit{ad}}\left({\overrightarrow{e}}_b{g}_{\mathit{cd}}+{\overrightarrow{e}}_c{g}_{\mathit{bd}}-{\overrightarrow{e}}_d{g}_{\mathit{bc}}\right),{\psi }_a^b=g^{\mathit{bc}}{\omega }_{\mathit{ac}},C_{\mathit{ab}}=\frac{1}{2}{\partial }_n{g}_{\mathit{ab}},C_a^b=g^{\mathit{bc}}{C}_{\mathit{ac}}.\end{array}$$E7

Here the endomorphism $\psi :\mathit{TM}\to \mathit{TM}$ is determined by the equality $\omega \left(X,Y\right)=g\left(\mathit{\psi X},Y\right)$. The following relations also hold: $C\left(X,Y\right)=\frac{1}{2}\left(L_{\overrightarrow{\xi }}g\right)\left(X,Y\right)$, $g\left(\mathit{CX},Y\right)=C\left(X,Y\right)$.

Note that in the case when $\omega \left(\overrightarrow{\xi },\cdot \right)=0$, the expressions with respect to the adapted coordinates for the Christoffel symbols ${\tilde{\mathrm{\Gamma }}}_{\mathit{ij}}^k$ of the Levi-Civita connection of the sub-Riemannian manifold take a simpler form. The following Christoffel symbols remain non-zero:

$${\tilde{\mathrm{\Gamma }}}_{\mathit{ab}}^c={\mathrm{\Gamma }}_{\mathit{ab}}^c,{\tilde{\mathrm{\Gamma }}}_{\mathit{ab}}^n={\omega }_{\mathit{ba}}-C_{\mathit{ab}},{\tilde{\mathrm{\Gamma }}}_{\mathit{\text{an}}}^b={\tilde{\mathrm{\Gamma }}}_{\mathit{na}}^b=C_a^b+{\psi }_a^b.$$E8

Let $\nabla$ be an intrinsic linear connection, let ${\nabla }^N$ be a connection uniquely determined by the conditions (1) ${\nabla }_X^{N}Y\in \mathrm{\Gamma }\left(\mathrm{D}\right)$, (2) ${\nabla }_X^N\overrightarrow{\xi }=\overrightarrow{0}$, (3) ${\nabla }_{\overrightarrow{\xi }}^{N}Y=\left[\overrightarrow{\xi },Y\right]+\mathit{NY}$, (4) ${\nabla }_Y^{N}Z={\nabla }_{Y}Z$, $X\in \mathrm{\Gamma }\left(\mathit{TM}\right);Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right)$. Throughout the paper, we will assume that the torsion $T\left(X,Y\right)$ of the intrinsic linear connection $\nabla$ is equal to zero:

$$T\left(X,Y\right)={\nabla }_{X}Y-{\nabla }_{Y}X-P\left[X,Y\right]=0.$$

It can be checked directly that the torsion $S\left(X,Y\right)$ of the extended connection ${\nabla }^N$ may be found by the following formula [15]:

$$S\left(X,Y\right)=2\omega \left(X,Y\right)\overrightarrow{\xi }+\eta \left(X\right)\mathit{NY}-\eta \left(Y\right)\mathit{NX},X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right).$$E9

Here

$$\omega \left(X,Y\right)=\mathit{d\eta }\left(X,Y\right).$$

If ${\nabla }_{X}g\left(Y,Z\right)=0$, $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right)$, then the following equality holds true:

$${\nabla }_X^{N}Y={\tilde{\nabla }}_{X}Y+\left({\tilde{\nabla }}_X\eta \right)\left(Y\right)\overrightarrow{\xi }-\eta \left(Y\right){\tilde{\nabla }}_X\overrightarrow{\xi }-\eta \left(X\right)\left(C+\psi -N\right)Y.$$E10

It can be directly verified that in this case the non-zero Christoffel symbols $G_{\mathit{jk}}^i$ of the connection ${\nabla }^N$ with respect to the adapted coordinates have the form

$$G_{\mathit{bc}}^a=\frac{1}{2}{g}^{\mathit{ad}}\left({\overrightarrow{e}}_b{g}_{\mathit{cd}}+{\overrightarrow{e}}_c{g}_{\mathit{bd}}-{\overrightarrow{e}}_d{g}_{\mathit{bc}}\right),{\mathrm{G}}_{\mathit{na}}^b=N_a^b.$$E11

In the case of intrinsic linear connection (not necessarily metric), the coefficients $G_{\mathit{bc}}^a$ may be found from the relation ${\nabla }_a{\overrightarrow{e}}_b=G_{\mathit{ab}}^c{\overrightarrow{e}}_c$.

Let $\tilde{S}\left(X,Y,Z\right)=g\left(S\left(X,Y\right),Z\right)$, $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right)$. With respect to the adapted coordinates, the non-zero components of the tensor $\tilde{S}\left(X,Y,Z\right)=g\left(S\left(X,Y\right),Z\right)$, $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right)$, have the following form:

$$\begin{array}{l}\tilde{S}\left({\overrightarrow{\mathrm{e}}}_{\mathrm{a}},{\overrightarrow{e}}_b,{\partial }_n\right)=2{\omega }_{\mathit{ab}},\\ \tilde{S}\left({\overrightarrow{\mathrm{e}}}_{\mathrm{a}},{\partial }_n,{\overrightarrow{e}}_b\right)=-g\left(N{\overrightarrow{\mathrm{e}}}_{\mathrm{a}},{\overrightarrow{e}}_b\right),\\ \tilde{S}\left({\partial }_n,{\overrightarrow{e}}_a,{\overrightarrow{e}}_b\right)=g\left(N{\overrightarrow{\mathrm{e}}}_{\mathrm{a}},{\overrightarrow{e}}_b\right).\end{array}$$

The concept of a quasi-statistical structure on a Riemannian manifold is discussed in [1]. A triplet $\left(M,g,\nabla \right)$ is called a quasi-statistical structure if the following equality holds [1]:

$${\nabla }_{X}g\left(Y,Z\right)-{\nabla }_{Y}g\left(X,Z\right)+\tilde{S}\left(X,Y,Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right).$$E12

We will call the triplet $\left(M,g,{\nabla }^N\right)$ a sub-Riemannian quasi-statistical structure (SRCS-structure) if the equality

$$\mathrm{\Phi }\left(X,Y,Z\right)={\nabla }_X^{N}g\left(Y,Z\right)-{\nabla }_Y^{N}g\left(X,Z\right)+\tilde{S}\left(X,Y,Z\right)-2\omega \left(X,Y\right)\eta \left(Z\right)=0,$$

where $\tilde{S}\left(X,Y,Z\right)=g\left(S\left(X,Y\right),Z\right)$, $X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right)$, $\omega =\mathit{d\eta }$, holds true. In this case, we will assume that the torsion of the intrinsic connection $\nabla$ is equal to zero. Note that if $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-statistical structure, then

$${\nabla }_{X}g\left(Y,Z\right)-{\nabla }_{Y}g\left(X,Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right).$$E13

We claim that if the triplet $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-statistical structure, then the corresponding intrinsic connection $\nabla$ is compatible with the metric $g$ restricted to the distribution $D$.

Theorem 1. A triple $\left(M,g,{\nabla }^N\right)$ is an SRCS-structure of a sub-Riemannian manifold $M$ of contact type if and only if $N=2C$ and

$${\nabla }_{X}g\left(Y,Z\right)-{\nabla }_{Y}g\left(X,Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right).$$E14

Proof.

Let us consider the equality

$${\nabla }_X^{N}g\left(Y,Z\right)-{\nabla }_Y^{N}g\left(X,Z\right)+\tilde{S}\left(X,Y,Z\right)-2\omega \left(X,Y\right)\eta \left(Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right)$$E15

We fix adapted coordinates. If $X={\partial }_n,Y={\overrightarrow{e}}_b,Z={\overrightarrow{e}}_c$, then we get

$${\nabla }_n^{N}g\left({\overrightarrow{e}}_b,{\overrightarrow{e}}_c\right)+g\left(N{\overrightarrow{\mathrm{e}}}_{\mathrm{b}},{\overrightarrow{e}}_c\right)=0.$$

Hence it holds

$${\partial }_n{g}_{\mathit{bc}}-N_b^d{g}_{\mathit{da}}-N_c^d{g}_{\mathit{bd}}+N_b^d{g}_{\mathit{da}}=0.$$

Finally, we get the equality

$$N_b^a=2C_b^a.$$E16

This proves the theorem.

Consider, as an example, a non-holonomic Kenmotsu manifold. The Kenmotsu manifolds were discovered in 1972 in [18]. The concept of a non-holonomic Kenmotsu manifold was introduced in [19]. In [20], a non-holonomic Kenmotsu manifold is equipped with an N-connection ${\nabla }^N$ with torsion of a special type. A normal almost contact metric manifold $M$ is called a non-holonomic Kenmotsu manifold if $L_{\overrightarrow{\xi }}g=2\left(g-\eta \otimes \eta \right)$ holds true [19].

Let now $\left(M,g,{\nabla }^N\right)$ be an SRCS-structure. Turning to the equation

$${\nabla }_X^{N}g\left(Y,Z\right)-{\nabla }_Y^{N}g\left(X,Z\right)+\tilde{S}\left(X,Y,Z\right)-2\omega \left(X,Y\right)\eta \left(Z\right)=0,$$E17

we obtain the following modification of Theorem 1.

Theorem 2. If $\nabla$ is an intrinsic metric connection, then the triple $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-statistical structure on a non-holonomic Kenmotsu manifold $M$ if and only if $N=2\tilde{E}$, where $\tilde{E}X=X,\tilde{E}\overrightarrow{\xi }=\overrightarrow{0}$, $X\in \mathrm{\Gamma }\left(\mathrm{D}\right)$.

Thus with respect to adapted coordinates, the non-zero Christoffel symbols $G_{\mathit{jk}}^i$ of the connection ${\nabla }^N$ in the case under consideration have the form

$$G_{\mathit{bc}}^a=\frac{1}{2}{g}^{\mathit{ad}}\left({\overrightarrow{e}}_b{g}_{\mathit{cd}}+{\overrightarrow{e}}_c{g}_{\mathit{bd}}-{\overrightarrow{e}}_d{g}_{\mathit{bc}}\right),{\mathrm{G}}_{\mathit{na}}^b=2{\delta }_b^a.$$E18

Let ${\tilde{\nabla }}^N$ be the connection conjugate to the connection ${\nabla }^N$ included in the SRCS-structure $\left(M,g,{\nabla }^N\right)$:

$$\mathit{Xg}\left(Y,Z\right)=g\left({\nabla }_X^{N}Y,Z\right)+g\left(Y,{\tilde{\nabla }}_X^{N}Z\right).$$E19

Denote by ${\tilde{\mathrm{G}}}_{\mathit{jk}}^i$ the Christoffel symbols of the connection ${\tilde{\nabla }}^N$ with respect to adapted coordinates. For $X={\overrightarrow{e}}_a,Y={\overrightarrow{e}}_b,Z={\overrightarrow{e}}_c$, we get

$${\tilde{\mathrm{G}}}_{\mathit{bc}}^a=G_{\mathit{bc}}^a.$$E20

In the case when $X={\partial }_n,Y={\overrightarrow{e}}_b,Z={\overrightarrow{e}}_c$, we have

$${\tilde{\mathrm{G}}}_{\mathit{na}}^b=0.$$E21

All other Christoffel symbols ${\tilde{\mathrm{G}}}_{\mathit{jk}}^i$ are zero.

Proposition 2. The non-zero Christoffel symbols ${\tilde{\mathrm{G}}}_{\mathit{jk}}^i$ of the connection ${\tilde{\nabla }}^N$ have the form: ${\tilde{\mathrm{G}}}_{\mathit{bc}}^a=G_{\mathit{bc}}^a$. In the case of a metric connection $\nabla$ it holds

$${\tilde{\mathrm{G}}}_{\mathit{bc}}^a=\frac{1}{2}{g}^{\mathit{ad}}\left({\overrightarrow{e}}_b{g}_{\mathit{cd}}+{\overrightarrow{e}}_c{g}_{\mathit{bd}}-{\overrightarrow{e}}_d{g}_{\mathit{bc}}\right).$$E22

Thus, it is shown that the conjugate connection to a connection ${\nabla }^N$ also has the structure of an N-connection with zero structural endomorphism $N$.

The following statement is an analogue of Proposition 1 from [6].

Proposition 3. Let $Q$ be a tensor field of type (1, 2) such that $g\left(Q\left(X,Z\right),Y\right)=g\left(X,Q\left(Y,Z\right)\right)$. Then $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-statistical structure if and only if $\left(M,g,{\dot{\nabla }}^N={\nabla }^N+Q\right)$ is a sub-Riemannian quasi-statistical structure.

Proof. Let us use the equalities obtained in [6],

$$\dot{S}\left(X,Y\right)=S\left(X,Y\right)+Q\left(X,Y\right)-Q\left(Y,X\right),$$E23

$${\dot{\nabla }}_X^{N}g\left(Y,Z\right)={\nabla }_X^{N}g\left(Y,Z\right)-g\left(Q\left(X,Y\right),Z\right)-g\left(Y,Q\left(X,Z\right)\right).$$E24

This implies

$$\dot{\mathrm{\Phi }}\left(X,Y,Z\right)=\mathrm{\Phi }\left(X,Y,Z\right)+g\left(X,Q\left(Y,Z\right)\right)-g\left(Y,Q\left(X,Z\right)\right),X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right).$$E25

Here

$$\dot{\mathrm{\Phi }}\left(X,Y,Z\right)={\dot{\nabla }}_X^{N}g\left(Y,Z\right)-{\dot{\nabla }}_Y^{N}g\left(X,Z\right)+\tilde{\dot{S}}\left(X,Y,Z\right)-2\omega \left(X,Y\right)\eta \left(Z\right)=0.$$E26

Thus the proposition is proved.

Below is the definition of a quasi-semi-Weyl structure adapted to our case, first published in [6]. A triple $\left(M,g,{\nabla }^N\right)$ is called a quasi-semi-Weyl structure if the following equality holds:

$${\nabla }_X^{N}g\left(Y,Z\right)+\eta \left(X\right)g\left(Y,Z\right)={\nabla }_Y^{N}g\left(X,Z\right)+\eta \left(Y\right)g\left(X,Z\right)-\tilde{S}\left(X,Y,Z\right).$$E27

We rewrite the last equality with respect to adapted coordinates:

1. If $X={\overrightarrow{e}}_a,Y={\overrightarrow{e}}_b,Z={\partial }_n$, then it follows that $\mathit{d\eta }=0$;

2. If $X={\partial }_n,Y={\overrightarrow{e}}_b,Z={\overrightarrow{e}}_c$, then we get:

$${\nabla }_n^{N}g\left({\overrightarrow{e}}_b,{\overrightarrow{e}}_c\right)+g\left({\overrightarrow{e}}_b,{\overrightarrow{e}}_c\right)=-g\left(N{\overrightarrow{\mathrm{e}}}_{\mathrm{b}},{\overrightarrow{e}}_c\right).$$

Hence with respect to adapted coordinates we have

$${\partial }_n{g}_{\mathit{bc}}-N_b^d{g}_{\mathit{da}}-N_c^d{g}_{\mathit{bd}}+g_{\mathit{bc}}=-N_b^d{g}_{\mathit{da}}.$$E28

Finally, we get the equality

$$N_b^a=2C_b^a+{\delta }_b^a.$$E29

Thus the following theorem holds true.

Theorem 3. A triple $\left(M,g,{\nabla }^N\right)$ is a quasi-semi-Weyl structure on a sub-Riemannian $M$ manifold of contact type if and only if the following conditions are satisfied:

1. The distribution $D$ of the manifold $M$ is involutive;

2. Structural endomorphism $N$ is of the following form:

   $$N=2C+\tilde{E},\text{where},\tilde{E}X=X,\tilde{E}\overrightarrow{\xi }=\overrightarrow{0},X\in \mathrm{\Gamma }\left(\mathrm{D}\right);$$

3. The intrinsic connection $\nabla$ is compatible with the metric $g$ restricted to the distribution $D$

$${\nabla }_{X}g\left(Y,Z\right)-{\nabla }_{Y}g\left(X,Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right).$$E30

In order to abandon the requirement that the distribution $D$ of the manifold $M$ is involutive, we refine the definition of a quasi-semi-Weyl structure for the case of a sub-Riemannian manifold of contact type. Namely, let us call the triple $\left(M,g,{\nabla }^N\right)$ a sub-Riemannian quasi-semi-Weyl structure if the following equality holds

$$\begin{array}{l}{\nabla }_X^{N}g\left(Y,Z\right)+\eta \left(X\right)g\left(Y,Z\right)={\nabla }_Y^{N}g\left(X,Z\right)+\eta \left(Y\right)g\left(X,Z\right)-\tilde{S}\left(X,Y,Z\right)+2\omega \left(X,Y\right)\eta \left(Z\right),X,\\ Y,Z\in \mathrm{\Gamma }\left(\mathrm{TM}\right).\end{array}$$E31

Let us rewrite the last equality with respect to adapted coordinates.

If $X={\partial }_n,Y={\overrightarrow{e}}_b,Z={\overrightarrow{e}}_c$, then we get:

$${\nabla }_n^{N}g\left({\overrightarrow{e}}_b,{\overrightarrow{e}}_c\right)+g\left({\overrightarrow{e}}_b,{\overrightarrow{e}}_c\right)=-g\left(N{\overrightarrow{\mathrm{e}}}_{\mathrm{b}},{\overrightarrow{e}}_c\right).$$E32

Hence with respect to adapted coordinates it holds that

$${\partial }_n{g}_{\mathit{bc}}-N_b^d{g}_{\mathit{da}}-N_c^d{g}_{\mathit{bd}}+g_{\mathit{bc}}=-N_b^d{g}_{\mathit{da}}.$$E33

Finally, we get the equality

$$N_b^a=2C_b^a+{\delta }_b^a.$$E34

Thus the following theorem holds true.

Theorem 4. A triple $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-semi-Weyl structure of a sub-Riemannian manifold $M$ of contact type if and only if

$$1.N=2C+\tilde{E},\text{where}\tilde{E}X=X,\tilde{E}\overrightarrow{\xi }=\overrightarrow{0},X\in \mathrm{\Gamma }\left(\mathrm{D}\right);$$E35

$$2.{\nabla }_{X}g\left(Y,Z\right)-{\nabla }_{Y}g\left(X,Z\right)=0,X,Y,Z\in \mathrm{\Gamma }\left(\mathrm{D}\right).$$E36

The following theorem establishes connections between the concepts of a sub-Riemannian quasi-statistical structure and a sub-Riemannian quasi-semi-Weyl structure of a sub-Riemannian manifold $M$ of contact type. It is a consequence of Theorems 1 and 4.

Theorem 5. Let the connections ${\nabla }^N$ and ${\nabla }^{\dot{N}}$ be extensions of the same intrinsic connection $\nabla$ on a sub-Riemannian manifold $M$ of contact type. Then the triple $\left(M,g,{\nabla }^N\right)$ is a sub-Riemannian quasi-statistical structure if and only if the triple $\left(M,g,{\nabla }^{\dot{N}}\right)$ is a sub-Riemannian quasi-semi-Weyl structure with

$$\dot{N}=N+\tilde{E}$$E37

3. Conclusion

This chapter makes an additional contribution to the theory of N-connections used for the development and numerous applications of contact-type sub-Riemannian geometry. The connection ${\nabla }^N$, which is an extension of the intrinsic connection $\nabla$ through endomorphism $N$, in a certain sense organically corresponds to the ideas of geometrization of a vast class of problems in theoretical physics. Above have spoken about the contribution of the theory of N-connections to the development of the geometry of Einstein manifolds and their generalizations [7]. The role of N-connections in the well-known geometric interpretation of the motion of a charged particle in the unified theory of gravitational and electromagnetic interactions has yet to be essentially clarified [9, 21].