Invariant Geometric Objects of the Equitorsion Canonical Biholomorphically Projective Mappings of Generalized Riemannian Space in the Eisenhart Sense

Abstract

The study of the equitorsion biholomorphically projective mappings between two generalized Riemannian spaces in the sense of Eisenhart’s definition is continued. Some new invariant geometric objects of an equitorsion canonical biholomorphically projective mapping are found, as well as some relations between these objects. At the end, the linear independence of the obtained invariants is examined.

1. Introduction

Differentiable manifolds ${\mathbb{GR}}_N$ with a non-symmetric metric tensor, ${\mathbb{GA}}_N$, and non-symmetric affine connection and their mappings are of interest to many scientists [1,2,3,4,5,6,7,8,9,10]. Although the notion of non-symmetric affine connection was used in several works before A. Einstein, for example in [11], the use of non-symmetric connection became especially evident after the appearance of the works of A. Einstein, relating to create his Unified Field Theory [12,13], where the symmetric part of a basic tensor is related to gravitation and the anti-symmetric one to electromagnetism. Since 1951, Eisenhart was occupied with problems of spaces with non-symmetric basic tensors and non-symmetric connection in several works [14]. He defined a generalized Riemannian space as “a space of coordinates with which ia associated a non-symmetric tensor $g_{ij}$”, and the connection coefficients are defined by $g_{ij}.$ In 1952, Eisenhart obtained two curvature tensors in generalized Riemannian space, using the fact that the connection coefficients are non-symmetric.

Geometric mappings are interesting and significant, both theoretically and practically. Geodesic and almost geodesic lines are important for geometry and physics. Geodesic mappings and their generalizations, like F-planar and holomorphically projective mappings, have been considered in many papers [1,4,6,9,10,15,16,17]. As a kind of generalization of holomorphically projective mappings, biholomorphically projective mappings were defined and considered in the paper [18].

In the sense of Eisenhart’s definition, an N-dimensional manifold ${\mathcal{M}}_N$ equipped with a non-symmetric metric tensor $\widehat{g}$, whose components are $g_{ij}$ such that $g_{ij}\ne g_{ji}$ for at least one pair of indices $(i,j)$, is a generalized (pseudo-)Riemannian space ${\mathbb{GR}}_N$. The symmetric and anti-symmetric parts of the metric $\widehat{g}$ are

$$\begin{array}{ccc}{g}_{\underline{ij}}={\displaystyle \frac{1}{2}}\left(g_{ij}+g_{ji}\right)& \mathrm{and}& g_{\underset{\vee }{ij}}={\displaystyle \frac{1}{2}}\left(g_{ij}-g_{ji}\right).\end{array}$$

(1)

We also assume that $det\left[g_{\underline{ij}}\right]\ne 0$, which we use to define a contravariant symmetric metric tensor by components such as $\left[g^{\underline{ij}}\right]={\left[g_{\underline{ij}}\right]}^{-1}$. The affine connection coefficients of the space ${\mathbb{GR}}_N$ are generalized Christoffell symbols ${\Gamma }_{jk}^i$ given by [18],

$$\begin{array}{c}\hfill {\Gamma }_{jk}^i=g^{\underline{ip}}{\Gamma }_{p.jk},\end{array}$$

where ${\Gamma }_{i.jk}={\displaystyle \frac{1}{2}}(g_{ji,k}-g_{jk,i}+g_{ik,j}),$ where, for example, $g_{ij,k}={\displaystyle \frac{\partial g_{ij}}{\partial x^k}}$. Generally, ${\Gamma }_{jk}^i\ne {\Gamma }_{kj}^i$, and the symmetric and anti-symmetric parts of ${\Gamma }_{jk}^i$, respectively, are given by the formulas

$${\Gamma }_{\underline{jk}}^i={\displaystyle \frac{1}{2}}({\Gamma }_{jk}^i+{\Gamma }_{kj}^i),{\Gamma }_{\underset{\vee }{jk}}^i={\displaystyle \frac{1}{2}}({\Gamma }_{jk}^i-{\Gamma }_{kj}^i).$$

The magnitude ${\Gamma }_{\underset{\vee }{jk}}^i$ is the torsion tensor of the space ${\mathbb{GR}}_N$.

In the case of generalized Riemannian space ${\mathbb{GR}}_N$, the following curvature tensors are linearly independent [19]:

$$\begin{array}{c}{\underset{1}{R}}_{jmn}^i=R_{jmn}^i+{\Gamma }_{\underset{\vee }{jm}|n}^i-{\Gamma }_{\underset{\vee }{jn}|m}^i+{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i,\hfill \end{array}$$

(2)

$$\begin{array}{c}{\underset{2}{R}}_{jmn}^i=R_{jmn}^i-{\Gamma }_{\underset{\vee }{jm}|n}^i+{\Gamma }_{\underset{\vee }{jn}|m}^i+{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i,\hfill \end{array}$$

(3)

$$\begin{array}{c}{\underset{3}{R}}_{jmn}^i=R_{jmn}^i+{\Gamma }_{\underset{\vee }{jm}|n}^i+{\Gamma }_{\underset{\vee }{jn}|m}^i-{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-2{\Gamma }_{\underset{\vee }{mn}}^p{\Gamma }_{\underset{\vee }{pj}}^i,\hfill \end{array}$$

(4)

$$\begin{array}{c}{\underset{4}{R}}_{jmn}^i=R_{jmn}^i+{\Gamma }_{\underset{\vee }{jm}|n}^i+{\Gamma }_{\underset{\vee }{jn}|m}^i-{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i+2{\Gamma }_{\underset{\vee }{mn}}^p{\Gamma }_{\underset{\vee }{pj}}^i,\hfill \end{array}$$

(5)

$$\begin{array}{c}{\underset{5}{R}}_{jmn}^i=R_{jmn}^i-{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i,\hfill \end{array}$$

(6)

$$\begin{array}{c}{\underset{6}{R}}_{jmn}^i=R_{jmn}^i+{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i,\hfill \end{array}$$

(7)

where “|” denotes a covariant derivative with respect to the symmetric affine connection ${\Gamma }_{\underline{jk}}^i$, whose curvature tensor is $R_{jmn}^i$.

The family of curvature tensors of the space ${\mathbb{GR}}_N$ is

$$K_{jmn}^i=R_{jmn}^i+u{\Gamma }_{\underset{\vee }{jm}|n}^i+u^{\prime }{\Gamma }_{\underset{\vee }{jn}|m}^i+v{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+v^{\prime }{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i+w{\Gamma }_{\underset{\vee }{mn}}^p{\Gamma }_{\underset{\vee }{pj}}^i,$$

(8)

for scalars u, $u^{\prime }$, v, $v^{\prime }$, and w.

The linearly independent curvature tensors (2)–(7) are extracted from the family (8) by selecting the following constants: $(u,u^{\prime },v,v^{\prime },w)$:

$$\begin{array}{ll}(u,u^{\prime },v,v^{\prime },w)\in \{& (1,-1,1,-1,0),(-1,1,1,-1,0),(1,1,-1,1,-2),\\ & (1,1,-1,1,2),(1,0,0,-1,1,0),(1,0,0,1,1,0)\}.\end{array}$$

(9)

Remark 1.

Let us obtain invariants for a geodesic mapping $f:{\mathbb{R}}_N\to {\overline{\mathbb{R}}}_N$ in the manner proposed by H. Weyl [20]. The basic equation of this mapping is

$${\overline{\Gamma }}_{\underline{jk}}^i={\Gamma }_{\underline{jk}}^i+{\psi }_k{\delta }_j^i+{\psi }_j{\delta }_k^i.$$

(10)

From this basic equation, one finds the transformation rule of curvature tensor $R_{jmn}^i$ to ${\overline{R}}_{jmn}^i$,

$${\overline{R}}_{jmn}^i=R_{jmn}^i+{\delta }_j^i\left({\psi }_{m|n}-{\psi }_{n|m}\right)+{\delta }_m^i{\psi }_{j|n}-{\delta }_n^i{\psi }_{j|m}-{\delta }_m^i{\psi }_j{\psi }_n+{\delta }_n^i{\psi }_j{\psi }_m.$$

(11)

If the exchange ${\psi }_{ij}={\psi }_{i|j}-{\psi }_i{\psi }_j$ is involved in Equation (11), we will transform it to

$${\overline{R}}_{jmn}^i=R_{jmn}^i+{\delta }_j^i\left({\psi }_{mn}-{\psi }_{nm}\right)+{\delta }_m^i{\psi }_{jn}-{\delta }_n^i{\psi }_{jm}.$$

(12)

After contracting the relation (12) by i and j, and by i and m, and with respect to the symmetry $R_{ij}=R_{ji}$ in Riemannian space, such as $R_{pmn}^p=R_{nm}-R_{mn}$, we obtain

$${\psi }_{ij}={\psi }_{ji}=-{\displaystyle \frac{1}{N-1}}\left({\overline{R}}_{ij}-R_{ij}\right).$$

(13)

After substituting the expression (13) into (12), we obtain

$${\overline{R}}_{jmn}^i=R_{jmn}^i-{\displaystyle \frac{1}{N-1}}\left({\delta }_m^i{\overline{R}}_{jn}-{\delta }_n^i{\overline{R}}_{jm}\right)+{\displaystyle \frac{1}{N-1}}\left({\delta }_m^i{R}_{jn}-{\delta }_n^i{R}_{jm}\right),$$

(14)

From Equation (14), we obtain that the Weyl projective tensor

$$W_{jmn}^i=R_{jmn}^i+{\displaystyle \frac{1}{N-1}}\left({\delta }_m^i{R}_{jn}-{\delta }_n^i{R}_{jm}\right)$$

is an invariant for the geodesic mapping f. The traces of $W_{jmn}^i$ are $W_{pmn}^p=0$, $W_{jpn}^p=0$, and $W_{jmp}^p=0$.

For this reason, there are two problems:

1. 

We do not have an invariant for the geodesic mapping that does not neglect the Ricci tensor $R_{ij}$ after contraction.

2. 

If there are not Kronecker delta-symbols in the transformation rule of the curvature tensor under an analyzed mapping, we are not able to apply Weyl’s method to obtain an invariant for the studied mapping.

These problems were solved in the paper [21] by obtaining general formulas for basic and derived invariants of a mapping between affine connection spaces. We will apply this methodology in this research and obtain invariants for a special mapping of generalized Riemannian space. Because the corresponding deformation tensor will not be a function of a Kronecker delta-symbol, we will be able to obtain just one invariant for the studied mapping, which is a function of the curvature tensor $R_{jmn}^i$. Unlike the Weyl projective tensor, the traces of the invariant that will be obtained in this paper will not vanish.

2. Review of Basic Invariants

One of the most important current problems in differential geometry is finding new invariant geometric objects. T. Y. Thomas [22] and H. Weyl [20] started the research about invariants of special diffeomorphisms between spaces with symmetric affine connection. Many authors have continued Thomas’s and Weyl’s works, for example V. Berezovski, J. Mikeš, E. Stepanova, etc. [1,2,5,15]. Also, some invariant geometrical objects for diffeomorphisms of spaces with non-symmetric affine connection are obtained in many papers, for example [9,16,17,23].

In this section, we will refer to the results presented in [21]. Namely, if $f:{\mathbb{R}}_N\to {\overline{\mathbb{R}}}_N$ is a mapping of Riemannian space ${\mathbb{R}}_N$ whose deformation tensor is

$${\overline{\Gamma }}_{\underline{jk}}^i={\Gamma }_{\underline{jk}}^i+{\overline{\omega }}_{jk}^i-{\omega }_{jk}^i,$$

(15)

for geometric objects ${\overline{\omega }}_{jk}^i$ and ${\omega }_{jk}^i$ symmetric by indices j and k, it is obtained that ${\overline{\mathcal{T}}}_{\underline{jk}}^i={\mathcal{T}}_{\underline{jk}}^i$ and ${\overline{\mathcal{W}}}_{jmn}^i={\mathcal{W}}_{jmn}^i$ for

$$\begin{array}{cc}\hfill & {\mathcal{T}}_{\underline{jk}}^i={\Gamma }_{\underline{jk}}^i+{\omega }_{jk}^i,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\mathcal{W}}_{jmn}^i=R_{jmn}^i+{\omega }_{jm|n}^i-{\omega }_{jn|m}^i+{\omega }_{jm}^p{\omega }_{pn}^i-{\omega }_{jn}^p{\omega }_{pm}^i,\hfill \end{array}$$

(17)

and the corresponding ${\overline{\mathcal{T}}}_{\underline{jk}}^i$ and ${\overline{\mathcal{W}}}_{jmn}^i$. The geometrical objects ${\mathcal{T}}_{\underline{jk}}^i$ and ${\mathcal{W}}_{jmn}^i$ are named the associated basic invariants of Thomas and Weyl type for the mapping f.

For an equitorsion mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$, the families curvature tensors (8) and

$${\overline{K}}_{jmn}^i={\overline{R}}_{jmn}^i+u{\overline{\Gamma }}_{\underset{\vee }{jm}\parallel n}^i+u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}\parallel m}^i+v{\overline{\Gamma }}_{\underset{\vee }{jm}}^p{\overline{\Gamma }}_{\underset{\vee }{pn}}^i+v^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}}^p{\overline{\Gamma }}_{\underset{\vee }{pm}}^i+w{\overline{\Gamma }}_{\underset{\vee }{mn}}^p{\overline{\Gamma }}_{\underset{\vee }{pj}}^i,$$

for the covariant derivatives with respect to torsion-free affine connections ${\Gamma }_{\underline{jk}}^i$ and ${\overline{\Gamma }}_{\underline{jk}}^i$, respectively, denoted by “|” and “‖”, satisfy the relation

$$\begin{array}{ll}{\overline{K}}_{jmn}^i-K_{jmn}^i& ={\overline{R}}_{jmn}^i-R_{jmn}^i+u\left({\overline{\Gamma }}_{\underline{pn}}^i{\overline{\Gamma }}_{\underset{\vee }{jm}}^p-{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p\right)-u\left({\overline{\Gamma }}_{\underline{jn}}^p{\overline{\Gamma }}_{\underset{\vee }{pm}}^i-{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i\right)\\ & -(u+u^{\prime })\left({\overline{\Gamma }}_{\underline{mn}}^p{\overline{\Gamma }}_{\underset{\vee }{jp}}^i-{\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i\right)+u^{\prime }\left({\overline{\Gamma }}_{\underline{pm}}^i{\overline{\Gamma }}_{\underset{\vee }{jn}}^p-{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p\right)\\ & -u^{\prime }\left({\overline{\Gamma }}_{\underline{jm}}^p{\overline{\Gamma }}_{\underset{\vee }{pn}}^i-{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i\right).\end{array}$$

(18)

Because the mapping f is equitorsion, the equality ${\overline{\Gamma }}_{\underset{\vee }{jk}}^i={\Gamma }_{\underset{\vee }{jk}}^i$ holds. With this in mind, and the invariance of geometrical objects ${\mathcal{T}}_{\underline{jk}}^i$ and ${\overline{\mathcal{W}}}_{jmn}^i$, the equality ${\widetilde{\overline{\mathcal{W}}}}_{jmn}^i={\tilde{\mathcal{W}}}_{jmn}^i$ is satisfied for

$$\begin{array}{cc}\hfill {\widetilde{\overline{\mathcal{W}}}}_{jmn}^i& ={\overline{\mathcal{W}}}_{jmn}^i+u{\overline{\mathcal{T}}}_{\underline{pn}}^i{\overline{\Gamma }}_{\underset{\vee }{jm}}^p-u{\overline{\mathcal{T}}}_{\underline{jn}}^p{\overline{\Gamma }}_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\overline{\mathcal{T}}}_{\underline{mn}}^p{\overline{\Gamma }}_{\underset{\vee }{jp}}^i+u^{\prime }{\overline{\mathcal{T}}}_{\underline{pm}}^i{\overline{\Gamma }}_{\underset{\vee }{jn}}^p-u^{\prime }{\overline{\mathcal{T}}}_{\underline{jm}}^p{\overline{\Gamma }}_{\underset{\vee }{pn}}^i,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\tilde{\mathcal{W}}}_{jmn}^i& ={\mathcal{W}}_{jmn}^i+u{\mathcal{T}}_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-u{\mathcal{T}}_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\mathcal{T}}_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+u^{\prime }{\mathcal{T}}_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-u^{\prime }{\mathcal{T}}_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i.\hfill \end{array}$$

(20)

After substituting the expressions of ${\mathcal{T}}_{\underline{jk}}^i$ and ${\mathcal{W}}_{jmn}^i$ given by (16) and (17) and the corresponding expressions of ${\overline{\mathcal{T}}}_{\underline{jk}}^i$ and ${\overline{\mathcal{W}}}_{jmn}^i$ into the equalities (19) and (20), respectively, one obtains

$$\begin{array}{ll}{\widetilde{\overline{\mathcal{W}}}}_{jmn}^i& ={\overline{R}}_{jmn}^i+{\overline{\omega }}_{jm\parallel n}^i-{\overline{\omega }}_{jn\parallel m}^i+{\overline{\omega }}_{jm}^p{\overline{\omega }}_{pn}^i-{\overline{\omega }}_{jn}^p{\overline{\omega }}_{pm}^i\\ \hfill & + u{\overline{\Gamma }}_{\underline{pn}}^i{\overline{\Gamma }}_{\underset{\vee }{jm}}^p-u{\overline{\Gamma }}_{\underline{jn}}^p{\overline{\Gamma }}_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\overline{\Gamma }}_{\underline{mn}}^p{\overline{\Gamma }}_{\underset{\vee }{jp}}^i+u^{\prime }{\overline{\Gamma }}_{\underline{pm}}^i{\overline{\Gamma }}_{\underset{\vee }{jn}}^p-u^{\prime }{\overline{\Gamma }}_{\underline{jm}}^p{\overline{\Gamma }}_{\underset{\vee }{pn}}^i\hfill \\ \hfill & + u{\overline{\Gamma }}_{\underset{\vee }{jm}}^p{\overline{\omega }}_{pn}^i-u{\overline{\Gamma }}_{\underset{\vee }{pm}}^i{\overline{\omega }}_{jn}^p-(u+u^{\prime }){\overline{\Gamma }}_{\underset{\vee }{jp}}^i{\overline{\omega }}_{mn}^p+u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}}^p{\overline{\omega }}_{pm}^i-u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{pn}}^i{\overline{\omega }}_{jm}^p,\hfill \end{array}$$

(21)

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{jmn}^i& =R_{jmn}^i+{\omega }_{jm|n}^i-{\omega }_{jn|m}^i+{\omega }_{jm}^p{\omega }_{pn}^i-{\omega }_{jn}^p{\omega }_{pm}^i\\ \hfill & + u{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-u{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+u^{\prime }{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-u^{\prime }{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i\hfill \\ \hfill & + u{\Gamma }_{\underset{\vee }{jm}}^p{\omega }_{pn}^i-u{\Gamma }_{\underset{\vee }{pm}}^i{\omega }_{jn}^p-(u+u^{\prime }){\Gamma }_{\underset{\vee }{jp}}^i{\omega }_{mn}^p+u^{\prime }{\Gamma }_{\underset{\vee }{jn}}^p{\omega }_{pm}^i-u^{\prime }{\Gamma }_{\underset{\vee }{pn}}^i{\omega }_{jm}^p.\hfill \end{array}$$

(22)

The difference of geometrical objects ${\widetilde{\overline{\mathcal{W}}}}_{jmn}^i$ and ${\tilde{\mathcal{W}}}_{jmn}^i$ given by Formulas (21) and (22) is

$$\begin{array}{ll}0={\widetilde{\overline{\mathcal{W}}}}_{jmn}^i-{\tilde{\mathcal{W}}}_{jmn}^i& = {\overline{R}}_{jmn}^i-R_{jmn}^i+u\left({\overline{\Gamma }}_{\underline{pn}}^i{\overline{\Gamma }}_{\underset{\vee }{jm}}^p-{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p\right)-u\left({\overline{\Gamma }}_{\underline{jn}}^p{\overline{\Gamma }}_{\underset{\vee }{pm}}^i-{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i\right)\\ & - (u+u^{\prime })\left({\overline{\Gamma }}_{\underline{mn}}^p{\overline{\Gamma }}_{\underset{\vee }{jp}}^i-{\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i\right)+u^{\prime }\left({\overline{\Gamma }}_{\underline{pm}}^i{\overline{\Gamma }}_{\underset{\vee }{jn}}^p-{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p\right)\\ & - u^{\prime }\left({\overline{\Gamma }}_{\underline{jm}}^p{\overline{\Gamma }}_{\underset{\vee }{pn}}^i-{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i\right)+{\overline{\omega }}_{j[m\parallel n]}^i-{\omega }_{j\left[m\right|n]}^i+{\overline{\omega }}_{j[m}^p{\overline{\omega }}_{pn]}^i-{\omega }_{j[m}^p{\omega }_{pn]}^i\\ & + u{\overline{\Gamma }}_{\underset{\vee }{jm}}^p{\overline{\omega }}_{pn}^i-u{\overline{\Gamma }}_{\underset{\vee }{pm}}^i{\overline{\omega }}_{jn}^p-(u+u^{\prime }){\overline{\Gamma }}_{\underset{\vee }{jp}}^i{\overline{\omega }}_{mn}^p+u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}}^p{\overline{\omega }}_{pm}^i-u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{pn}}^i{\overline{\omega }}_{jm}^p\\ & - u{\Gamma }_{\underset{\vee }{jm}}^p{\omega }_{pn}^i+u{\Gamma }_{\underset{\vee }{pm}}^i{\omega }_{jn}^p+(u+u^{\prime }){\Gamma }_{\underset{\vee }{jp}}^i{\omega }_{mn}^p-u^{\prime }{\Gamma }_{\underset{\vee }{jn}}^p{\omega }_{pm}^i+u^{\prime }{\Gamma }_{\underset{\vee }{pn}}^i{\omega }_{jm}^p,\end{array}$$

(23)

where $\left[mn\right]$ is an anti-symmetrization without division by indices m and n.

The first seven summands in Equation (23) coincide with the right side of equality (18). Hence, the forthcoming equality holds,

$$\begin{array}{ll}0={\widetilde{\overline{\mathcal{W}}}}_{jmn}^i-{\tilde{\mathcal{W}}}_{jmn}^i& ={\overline{K}}_{jmn}^i-K_{jmn}^i+{\overline{\omega }}_{j[m\parallel n]}^i-{\omega }_{j\left[m\right|n]}^i+{\overline{\omega }}_{j[m}^p{\overline{\omega }}_{pn]}^i-{\omega }_{j[m}^p{\omega }_{pn]}^i\\ & + u{\overline{\Gamma }}_{\underset{\vee }{jm}}^p{\overline{\omega }}_{pn}^i-u{\overline{\Gamma }}_{\underset{\vee }{pm}}^i{\overline{\omega }}_{jn}^p-(u+u^{\prime }){\overline{\Gamma }}_{\underset{\vee }{jp}}^i{\overline{\omega }}_{mn}^p+u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}}^p{\overline{\omega }}_{pm}^i-u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{pn}}^i{\overline{\omega }}_{jm}^p\\ & - u{\Gamma }_{\underset{\vee }{jm}}^p{\omega }_{pn}^i+u{\Gamma }_{\underset{\vee }{pm}}^i{\omega }_{jn}^p+(u+u^{\prime }){\Gamma }_{\underset{\vee }{jp}}^i{\omega }_{mn}^p-u^{\prime }{\Gamma }_{\underset{\vee }{jn}}^p{\omega }_{pm}^i+u^{\prime }{\Gamma }_{\underset{\vee }{pn}}^i{\omega }_{jm}^p,\end{array}$$

(24)

for the families $K_{jmn}^i$ and ${\overline{K}}_{jmn}^i$ of the curvature tensors of the spaces ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$.

From the equality (24), we obtain the expressions of the geometrical objects ${\tilde{\mathcal{W}}}_{jmn}^i$ and ${\widetilde{\overline{\mathcal{W}}}}_{jmn}^i$ as follows:

$$\begin{array}{ll}{\widetilde{\overline{\mathcal{W}}}}_{jmn}^i& = {\overline{K}}_{jmn}^i+{\overline{\omega }}_{jm\parallel n}^i-{\overline{\omega }}_{jn\parallel m}^i+{\overline{\omega }}_{jm}^p{\overline{\omega }}_{pn}^i-{\overline{\omega }}_{jn}^p{\overline{\omega }}_{pm}^i\\ \hfill & + u{\overline{\Gamma }}_{\underset{\vee }{jm}}^p{\overline{\omega }}_{pn}^i-u{\overline{\Gamma }}_{\underset{\vee }{pm}}^i{\overline{\omega }}_{jn}^p-(u+u^{\prime }){\overline{\Gamma }}_{\underset{\vee }{jp}}^i{\overline{\omega }}_{mn}^p+u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{jn}}^p{\overline{\omega }}_{pm}^i-u^{\prime }{\overline{\Gamma }}_{\underset{\vee }{pn}}^i{\overline{\omega }}_{jm}^p,\hfill \end{array}$$

(25)

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{jmn}^i\hfill & =K_{jmn}^i+{\omega }_{jm|n}^i-{\omega }_{jn|m}^i+{\omega }_{jm}^p{\omega }_{pn}^i-{\omega }_{jn}^p{\omega }_{pm}^i\\ \hfill & -u{\Gamma }_{\underset{\vee }{jm}}^p{\omega }_{pn}^i+u{\Gamma }_{\underset{\vee }{pm}}^i{\omega }_{jn}^p+(u+u^{\prime }){\Gamma }_{\underset{\vee }{jp}}^i{\omega }_{mn}^p-u^{\prime }{\Gamma }_{\underset{\vee }{jn}}^p{\omega }_{pm}^i+u^{\prime }{\Gamma }_{\underset{\vee }{pn}}^i{\omega }_{jm}^p.\hfill \end{array}$$

(26)

The next theorem holds.

Theorem 1.

Let $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ be an equitorsion mapping of a generalized Riemannian space ${\mathbb{GR}}_N$. The geometrical objects ${\tilde{\mathcal{W}}}_{jmn}^i$ given by Equation (22) are invariants for the mapping f expressed in terms of the curvature tensor of the associated space ${\mathbb{R}}_N$.

The geometrical objects ${\tilde{\mathcal{W}}}_{jmn}^i$ given by Equation (26) are invariants for the mapping f expressed in terms of the curvature tensors of the space ${\mathbb{GR}}_N$.

The geometrical objects ${\tilde{\mathcal{W}}}_{jmn}^i$ are equitorsion invariants for the mapping f.

Because there are six linearly independent curvature tensors [19], the next corollaries are satisfied.

Corollary 1.

Let $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ be an equitorsion mapping of Riemannian space ${\mathbb{GR}}_N$. Three of the invariants for this mapping, given by Equation (22), are linearly independent.

Corollary 2.

Let $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ be an equitorsion mapping of Riemannian space ${\mathbb{GR}}_N$. Six of the invariants for this mapping, given by Equation (26), are linearly independent.

Corollary 3.

Let $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ be an equitorsion mapping of Riemannian space ${\mathbb{GR}}_N$. The geometrical objects ${\tilde{\mathcal{W}}}_{ij}={\tilde{\mathcal{W}}}_{ijp}^p$, whose components are

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{ij}& = R_{ij}+{\omega }_{ij|p}^p-{\omega }_{ip|j}^p+{\omega }_{ij}^q{\omega }_{qp}^p-{\omega }_{iq}^p{\omega }_{jp}^q\\ & + u{\Gamma }_{\underline{pq}}^q{\Gamma }_{\underset{\vee }{ij}}^p+u{\Gamma }_{\underline{ip}}^q{\Gamma }_{\underset{\vee }{jq}}^p-(u+u^{\prime }){\Gamma }_{\underline{jp}}^q{\Gamma }_{\underset{\vee }{iq}}^p+u^{\prime }{\Gamma }_{\underline{jp}}^q{\Gamma }_{\underset{\vee }{iq}}^p\\ & + u{\Gamma }_{\underset{\vee }{ij}}^p{\omega }_{pq}^q+u{\Gamma }_{\underset{\vee }{jq}}^p{\omega }_{ip}^q-(u+u^{\prime }){\Gamma }_{\underset{\vee }{iq}}^p{\omega }_{jp}^q+u^{\prime }{\Gamma }_{\underset{\vee }{iq}}^p{\omega }_{jp}^q,\end{array}$$

(27)

are invariants for the mapping f.

Remark 2.

When we express the invariant ${\tilde{\mathcal{W}}}_{jmn}^i$ in terms of curvature tensors of the space ${\mathbb{GR}}_N$, we involve summands like ${\Gamma }_{\underset{\vee }{jk}}^i{\Gamma }_{\underset{\vee }{qr}}^p$. These summands are invariants for the equitorsion mapping f, and they make six linearly independent invariants. These summands are extracted from the invariant ${\tilde{\mathcal{W}}}_{jmn}^i$ expressed in terms of the curvature tensor of the associated space ${\mathbb{R}}_N$ and other non-invariant objects. For this reason, we will obtain invariants without including other invariants as separate summands.

3. Invariants for Equitorsion Canonical Biholomorphically Projective Mappings

Let ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$ be two generalized Riemannian spaces. We will observe these spaces in the common system of coordinates defined by the mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N.$ If ${\Gamma }_{jk}^i$ and ${\overline{\Gamma }}_{jk}^i$ are connection coefficients of the spaces ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$, respectively, then $P_{jk}^i={\overline{\Gamma }}_{jk}^i-{\Gamma }_{jk}^i$ is the deformation tensor of the connection for a mapping f [18].

In the paper [18], we defined biholomorphically projective mappings between two generalized Riemannian spaces ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$ with almost complex structures that are equal in a common system of coordinates defined by the mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$. We considered a generalized Riemannian space ${\mathbb{GR}}_N$ with a non-symmetric metric tensor $g_{ij}$ and almost complex structure $F_j^i$ such that $F_j^i\ne a{\delta }_j^i$, where a is a scalar invariant, and we defined a biholomorphically projective curve of the kind $\theta$$(\theta =1,2)$ and a biholomorphically projective mapping of the kind $\theta (\theta =1,2)$.

Definition 1 

([18]). In the space ${\mathbb{GR}}_N$, a curve l given in parametric form

$$x^i=x^i\left(t\right),(i=1,\cdots ,N),$$

is said to be biholomorphically projective of the kind θ $(\theta =1,2)$ if it satisfies the following equation:

$${{\lambda }^h}_{\underset{\theta }{|}p}\left(t\right){\lambda }^p\left(t\right)=a\left(t\right){\lambda }^h\left(t\right)+b\left(t\right)F_p^h{\lambda }^p\left(t\right)+c\left(t\right){\stackrel{2}{F}}_p^h{\lambda }^p\left(t\right),$$

where $a,$$b,c$ are functions of parameter t, where ${\lambda }^i={\displaystyle \frac{d{x}^i}{dt}},$ ${\stackrel{2}{F}}_p^h=F_q^h{F}_p^q.$

Definition 2 

([18]). A difeomorphism $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ is a biholomorphically projective mapping of the kind $\theta (\theta =1,2)$ if biholomorphically projective curves of the kind $\theta (\theta =1,2)$ of the space ${\mathbb{GR}}_N$ are mapped to the biholomorphically projective curves of the kind θ of the space $\mathbb{G}{\overline{\mathbb{R}}}_N$.

After defining, we concluded that the biholomorphically projective curves of the first kind and the biholomorphically projective curves of the second kind match, so we simply named them the biholomorphically projective curves, and the corresponding mappings are named biholomorphically projective mappings.

Also, the mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ is a biholomorphically projective mapping if in the common coordinate system the connection coefficients ${\Gamma }_{ij}^h$ and ${\overline{\Gamma }}_{ij}^h$ satisfy the relation [18]

$${\overline{\Gamma }}_{jk}^i={\Gamma }_{jk}^i+{\psi }_{(j}{\delta }_{k)}^i+{\sigma }_{(j}{F}_{k)}^i+{\tau }_{(j}{\stackrel{2}{F}}_{k)}^i+{\xi }_{jk}^i,$$

and the deformation tensor has the form [18]

$$P_{jk}^i={\psi }_{(j}{\delta }_{k)}^i+{\sigma }_{(j}{F}_{k)}^i+{\tau }_{(j}{\stackrel{2}{F}}_{k)}^i+{\xi }_{jk}^i,$$

(28)

where $\left(ij\right)$ is a symmetrization without division by indices $i,j$, ${\psi }_i$, ${\sigma }_i$, and ${\tau }_i$ are vectors, ${\stackrel{2}{F}}_j^i=F_q^i{F}_j^q,$ and ${\xi }_{jk}^i$ is an anti-symmetric tensor. Inspired by the form of the deformation tensor (28), we have defined another type of mapping.

Let $F_j^i$ and ${\overline{F}}_j^i$ be almost complex structures of the spaces ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$, respectively, where $F_j^i={\overline{F}}_j^i$ in the common system of coordinates defined by the mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$, and assume that it holds $F_j^i\ne a{\delta }_j^i,$ where a is a scalar invariant. The mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ is a canonical biholomorphically projective mapping if in the common coordinate system the connection coefficients ${\Gamma }_{jk}^i$ and ${\overline{\Gamma }}_{jk}^i$ satisfy the relation [24]

$${\overline{\Gamma }}_{jk}^i={\Gamma }_{jk}^i+{\sigma }_{(j}{F}_{k)}^i+{\tau }_{(j}{\stackrel{2}{F}}_{k)}^i+{\xi }_{jk}^i,$$

(29)

where $\left(jk\right)$ is a symmetrization without division by indices j and k, ${\sigma }_j$ and ${\tau }_j$ are vectors, ${\stackrel{2}{F}}_j^i=F_p^i{F}_j^p,$ and ${\xi }_{jk}^i$ is an anti-symmetric tensor. If $P_{jk}^i$ is a deformation tensor with respect to the canonical biholomorphically projective mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$, then we have [24]

$$P_{jk}^i={\sigma }_{(j}{F}_{k)}^i+{\tau }_{(j}{\stackrel{2}{F}}_{k)}^i+{\xi }_{jk}^i.$$

(30)

The mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ is an equitorsion canonical biholomorphically projective mapping if the torsion tensors of the spaces ${\mathbb{GR}}_N$ and $\mathbb{G}{\overline{\mathbb{R}}}_N$ are equal in a common coordinate system after the mapping f. Then, from Ref. [24],

$${\xi }_{jk}^i=0.$$

(31)

In this case, the relation (30) becomes [24]

$$P_{jk}^i={\sigma }_{(j}{F}_{k)}^i+{\tau }_{(j}{\stackrel{2}{F}}_{k)}^i.$$

(32)

As a special case of equitorsion canonical biholomorphically projective mappings, the equitorsion mapping $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$, whose deformation tensor is

$$P_{jk}^i={\overline{\Gamma }}_{jk}^i-{\Gamma }_{jk}^i=-{\tau }_p{F}_j^p{F}_k^i-{\tau }_p{F}_k^p{F}_j^i+{\tau }_j{\stackrel{2}{F}}_k^i+{\tau }_k{\stackrel{2}{F}}_j^i,$$

(33)

was studied. In this case, an invariant of Thomas type [24]

$${\mathcal{CHT}}_{jk}^i={\Gamma }_{\underline{jk}}^i-{\displaystyle \frac{1}{e}}\left({\Gamma }_{\underline{pq}}^q{F}_j^p{F}_k^i+{\Gamma }_{\underline{pq}}^q{F}_k^p{F}_j^i-{\Gamma }_{\underline{jp}}^p{\stackrel{2}{F}}_k^i-{\Gamma }_{\underline{kp}}^p{\stackrel{2}{F}}_j^i\right),$$

(34)

is obtained for $e=\pm 1$, assuming that ${\delta }_j^i$, $F_j^i$, and ${\stackrel{2}{F}}_j^i$ are linearly independent.

The invariant ${\mathcal{CHT}}_{jk}^i$ is the associated basic invariant for mapping f. From this invariant, one reads

$${\omega }_{jk}^i=-{\displaystyle \frac{1}{e}}\left({\Gamma }_{\underline{pq}}^q{F}_j^p{F}_k^i+{\Gamma }_{\underline{pq}}^q{F}_k^p{F}_j^i-{\Gamma }_{\underline{jp}}^p{\stackrel{2}{F}}_k^i-{\Gamma }_{\underline{kp}}^p{\stackrel{2}{F}}_j^i\right).$$

(35)

We will express the previous ${\omega }_{jk}^i$ in some other form, better for further computing

$${\omega }_{jk}^i=-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q\left(F_j^p{F}_k^i+F_k^p{F}_j^i-{\delta }_j^p{\stackrel{2}{F}}_k^i-{\delta }_k^p{\stackrel{2}{F}}_j^i\right)=-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{jk}^{pi},$$

(36)

for

$${\mathcal{F}}_{jk}^{pi}=F_j^p{F}_k^i+F_k^p{F}_j^i-{\delta }_j^p{\stackrel{2}{F}}_k^i-{\delta }_k^p{\stackrel{2}{F}}_j^i.$$

(37)

After substituting the expression (36) into Equations (17) and (22), assuming that F is covariantly constant, we obtain the following results:

$$\begin{array}{ccc}\hfill & {\mathcal{W}}_{jmn}^i\hfill & \hfill =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\end{array}$$

(38)

$$\begin{array}{ll}{\mathcal{W}}_{ij}\hfill & = R_{ij}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}|j}^q{\mathcal{F}}_{ir}^{pr}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}|r}^q{\mathcal{F}}_{ij}^{pr}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{ij|r}^{pr}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{ir|j}^{pr}\\ & +{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ij}^{ap}{\mathcal{F}}_{pr}^{br}-{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ir}^{ap}{\mathcal{F}}_{pj}^{br},\end{array}$$

(39)

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{jmn}^i\hfill & = R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\\ \hfill & + u{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-u{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+u^{\prime }{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-u^{\prime }{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}\hfill \\ \hfill & + {\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}+{\displaystyle \frac{u+u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{jp}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{mn}^{rp}-{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}+{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp},\hfill \end{array}$$

(40)

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{ij}\hfill & = R_{ij}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}|j}^q{\mathcal{F}}_{ir}^{pr}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}|r}^q{\mathcal{F}}_{ij}^{pr}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{ij|r}^{pr}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{ir|j}^{pr}\\ & + {\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ij}^{ap}{\mathcal{F}}_{pr}^{br}-{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ir}^{ap}{\mathcal{F}}_{pj}^{br}+u{\Gamma }_{\underline{pq}}^q{\Gamma }_{\underset{\vee }{ij}}^p+u{\Gamma }_{\underline{iq}}^p{\Gamma }_{\underset{\vee }{jp}}^q-u{\Gamma }_{\underline{jq}}^p{\Gamma }_{\underset{\vee }{ip}}^q\hfill \\ & - {\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{ij}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{ps}^{rs}+{\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{pj}}^s{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{is}^{rp}+{\displaystyle \frac{u+u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{ip}}^s{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{js}^{rp}-{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{is}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pj}^{rs}.\end{array}$$

(41)

Accordingly, we conclude that the following theorem is valid.

Theorem 2.

Let $f:{\mathbb{GR}}_N\to \mathbb{G}{\overline{\mathbb{R}}}_N$ be an equitorsion canonical biholomorphically projective mapping that preserves the structure F, which is covariantly constant according to the covariant derivative with respect to torsion-free affine connection. The geometrical objects ${\mathcal{W}}_{jmn}^i$ and ${\tilde{\mathcal{W}}}_{jmn}^i$, given by (38) and (40), respectively, are the associated basic invariant and equitorsion invariant for the mapping f.

Corollary 4.

The invariants ${\mathcal{W}}_{jmn}^i$ and ${\tilde{\mathcal{W}}}_{jmn}^i$ satisfy the equation

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{jmn}^i& = {\mathcal{W}}_{jmn}^i+u{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-u{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-(u+u^{\prime }){\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+u^{\prime }{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-u^{\prime }{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i\\ & - {\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}+{\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}+{\displaystyle \frac{u+u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{jp}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{mn}^{rp}-{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}+{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp}.\end{array}$$

(42)

The invariants ${\mathcal{W}}_{ij}$ and ${\tilde{\mathcal{W}}}_{ij}$ satisfy the equation

$$\begin{array}{ll}{\tilde{\mathcal{W}}}_{ij}& ={\mathcal{W}}_{ij}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ij}^{ap}{\mathcal{F}}_{pr}^{br}-{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{ir}^{ap}{\mathcal{F}}_{pj}^{br}+u{\Gamma }_{\underline{pq}}^q{\Gamma }_{\underset{\vee }{ij}}^p+u{\Gamma }_{\underline{iq}}^p{\Gamma }_{\underset{\vee }{jp}}^q-u{\Gamma }_{\underline{jq}}^p{\Gamma }_{\underset{\vee }{ip}}^q\\ & -{\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{ij}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{ps}^{rs}+{\displaystyle \frac{u}{e}}{\Gamma }_{\underset{\vee }{pj}}^s{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{is}^{rp}+{\displaystyle \frac{u+u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{ip}}^s{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{js}^{rp}-{\displaystyle \frac{u^{\prime }}{e}}{\Gamma }_{\underset{\vee }{is}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pj}^{rs}.\end{array}$$

(43)

The invariants ${\tilde{\mathcal{W}}}_{jmn}^i$, ${\mathcal{W}}_{jmn}^i$, ${\tilde{\mathcal{W}}}_{ij}$, and ${\mathcal{W}}_{ij}$ are tensors.

Linearly Independent Invariants

The family of invariants ${\tilde{\mathcal{W}}}_{jmn}^i$ given by (42) depends u and $u^{\prime }$, and none of the scalars present in the family of curvature tensors (8) feature. After changing the corresponding scalars u and $u^{\prime }$ from linearly independent curvature tensors into the Equation (41), we will obtain the following invariants:

$$\begin{array}{ll}{\underset{1}{\tilde{\mathcal{W}}}}_{jmn}^i\hfill & =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\\ \hfill & +{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p+{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}\hfill \\ \hfill & +{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp},\hfill \end{array}$$

(44)

$$\begin{array}{ll}{\underset{2}{\tilde{\mathcal{W}}}}_{jmn}^i\hfill & =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\\ \hfill & -{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p+{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i+{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}\hfill \\ \hfill & -{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp},\hfill \end{array}$$

(45)

$$\begin{array}{ll}{\underset{3}{\tilde{\mathcal{W}}}}_{jmn}^i\hfill & =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\\ \hfill & +{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-2{\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}\hfill \\ \hfill & +{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}+{\displaystyle \frac{2}{e}}{\Gamma }_{\underset{\vee }{jp}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{mn}^{rp}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp},\hfill \end{array}$$

(46)

$$\begin{array}{ll}{\underset{4}{\tilde{\mathcal{W}}}}_{jmn}^i& =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}\\ \hfill & +{\Gamma }_{\underline{pn}}^i{\Gamma }_{\underset{\vee }{jm}}^p-{\Gamma }_{\underline{jn}}^p{\Gamma }_{\underset{\vee }{pm}}^i-2{\Gamma }_{\underline{mn}}^p{\Gamma }_{\underset{\vee }{jp}}^i+{\Gamma }_{\underline{pm}}^i{\Gamma }_{\underset{\vee }{jn}}^p-{\Gamma }_{\underline{jm}}^p{\Gamma }_{\underset{\vee }{pn}}^i-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jm}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pn}^{ri}\hfill \\ \hfill & +{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pm}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jn}^{rp}+{\displaystyle \frac{2}{e}}{\Gamma }_{\underset{\vee }{jp}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{mn}^{rp}-{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{jn}}^p{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{pm}^{ri}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underset{\vee }{pn}}^i{\Gamma }_{\underline{rq}}^q{\mathcal{F}}_{jm}^{rp},\hfill \end{array}$$

(47)

$$\begin{array}{ll}{\underset{5}{\tilde{\mathcal{W}}}}_{jmn}^i\hfill & =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi},\end{array}$$

(48)

$$\begin{array}{cc}{\underset{6}{\tilde{\mathcal{W}}}}_{jmn}^i\hfill & \hfill =R_{jmn}^i+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}\left|\right[m}^q{\mathcal{F}}_{jn]}^{pi}+{\displaystyle \frac{1}{e}}{\Gamma }_{\underline{pq}}^q{\mathcal{F}}_{j\left[m\right|n]}^{pi}+{\Gamma }_{\underline{aq}}^q{\Gamma }_{\underline{bs}}^s{\mathcal{F}}_{j[m}^{ap}{\mathcal{F}}_{pn]}^{bi}.\end{array}$$

(49)

Theorem 3.

Three of the invariants (44)–(49) are linearly independent.

Proof. 

The family of invariants ${\tilde{\mathcal{W}}}_{jmn}^i$ given by (40) is isomorphic to the vector space, whose elements are $v=(1, u, -u, -u-u^{\prime }, u^{\prime }, -u^{\prime }, u, u, u+u^{\prime }, -u^{\prime }, u^{\prime })$. The fifths $(u, u^{\prime }, v, v^{\prime }, w)$, which generate six linearly independent curvature tensors, are given by (9). After taking the values of u and $u^{\prime }$ from these fifths and substituting them into the vectors v, we obtain the following vectors:

$$\begin{array}{cc}\hfill & v_1=(1,1,-1,0,-1,1,1,1,0,1,-1),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill & v_2=(1,-1,1,0,1,-1,-1,-1,0,-1,1),\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill & v_3=(1,1,-1,-2,1,-1,1,1,2,-1,1),\hfill \end{array}$$

(52)

$$\begin{array}{cc}\hfill & v_4=(1,1,-1,-2,1,-1,1,1,2,-1,1),\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & v_5=(1,0,0,0,0,0,0,0,0,0,0),\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill & v_6=(1,0,0,0,0,0,0,0,0,0,0).\hfill \end{array}$$

(55)

The rank of matrix $\left[\begin{array}{cccccc}{v}_1& v_2& v_3& v_4& v_5& v_6\end{array}\right]$ is 3, which completes the proof of this theorem. □

Corollary 5.

Three linearly independent invariants are ${\underset{1}{\tilde{\mathcal{W}}}}_{jmn}^i$, ${\underset{2}{\tilde{\mathcal{W}}}}_{jmn}^i$, and ${\underset{3}{\tilde{\mathcal{W}}}}_{jmn}^i$.

Corollary 6.

Three of the invariants ${\tilde{\mathcal{W}}}_{ij}$ are linearly independent. Invariants ${\underset{1}{\tilde{\mathcal{W}}}}_{ij}$, ${\underset{2}{\tilde{\mathcal{W}}}}_{ij}$, ${\underset{3}{\tilde{\mathcal{W}}}}_{ij}$ are linearly independent.

4. Discussion

Invariants for different mappings in the case of spaces with symmetric and non-symmetric affine connection have been obtained by different authors. The generalizations of the Weyl conformal, the Weyl projective tensor, and the Thomas projective parameters are objects that have been generalized in many different papers regarding invariants for different types of geometric mappings. This paper is a natural continuation of the research discussed in papers [21,24]. The study of the equitorsion biholomorphically projective mapping between two generalized Riemannian spaces in the sense of Eisenhart’s definition has been continued by finding new invariant geometric objects for the special case of the equitorsion canonical biholomorphically projective mapping. Also, the linear independence of the obtained invariants was examined. Considering the diversity of types of bihollomorphicaly projective mappings, the goal of further research will be to find new invariant geometric objects, both in the case of the considered equitorsion canonical biholomorphically projective mapping and in the case of some other types of biholomorphically projective mappings. The results of this paper motivate us to ask ourselves whether there is an interpretation from a physical point of view, as well as what the geometrical significance of the obtained objects is.