Conformal Equitorsion and Concircular Transformations in a Generalized Riemannian Space

Abstract

In the beginning, the basic facts about a conformal transformations are exposed and then equitorsion conformal transformations are defined. For every five independent curvature tensors in Generalized Riemannian space, the above cited transformations are investigated and corresponding invariants-5 concircular tensors of concircular transformations are found.

1. Introduction

In the sense of Eisenhart’s definition [1], a generalized Riemannian space $(G{R}_N)$ is a differentiable N-dimensional manifold that is endowed with basic non-symmetric tensor $(g_{ij}\ne g_{ji})$, where $det{g}_{ij}\ne 0.$

The symmetric part of $g_{ij}$ is noted with $g_{\underline{ij}}$ and antisymmetric one with $g_{\underset{V}{ij}}$. The lowering and rising of indices in $G{R}_N$ is defined by $g_{\underline{ij}}$ and $g^{\underline{ij}}$, respectively, where $g_{\underline{ij}}{g}^{\underline{ik}}={\delta }_j^k$ $(det{g}_{\underline{ij}}\ne 0)$. The Christoffel symbols in $G{R}_N$ are given in the next manner:

$$\begin{array}{c}a){\Gamma }_{i.jk}=\frac{1}{2}(g_{ji,k}-g_{jk,i}^{}+g_{ik,j}),b){\Gamma }_{jk}^i=g^{\underline{ip}}{\Gamma }_{p.jk}=\frac{1}{2}{g}^{\underline{ip}}(g_{jp,k}-g_{jk,p}+g_{pk,i}),\hfill \\ \mathrm{where},\mathrm{e}.\mathrm{g}.,g_{ij,k}=\partial g_{ij}/\partial x^k.\hfill \end{array}$$

(1)

Because of non-symmetry of the affine connection coefficients ${\Gamma }_{jk}^i$ by indices j and k, there are four kinds of covariant differentiation in the space ${\mathbb{GR}}_N$. Namely, for a tensor $a_j^i$, these covariant derivatives are defined as:

$$a_{j\underset{4}{\underset{3}{\underset{2}{\underset{1}{|}}}}m}^i=a_{j,m}^i+{\Gamma }_{\underset{mp}{\underset{pm}{\underset{mp}{\underset{}{pm}}}}}^i{a}_j^p-{\Gamma }_{\underset{jm}{\underset{mj}{\underset{mj}{jm}}}}^p{a}_p^i.$$

(2)

Yano in [2] investigates a conformal and concircular transformations in the $R_N$. In that case, of course, he considers one that is Riemannian curvature tensor. De and Mandal in [3] studied concircular curvature tensors as important tensors from the differential geometric point of view. In [4,5,6,7,8,9,10,11], Mikeš et al. have studied special kinds ot transformations in Riemannian space.

Minčić, in his doctoral dissertation (Novi Sad, 1975), obtained 12 curvature tensors, using non-symmetric connection. Among these 12 tensors, five of them are independent (se also [12,13,14,15,16,17]) and they are noted $\underset{1}{R},\dots ,\underset{5}{R}.$

In [18], another combination of five independent curvature tensors is obtained, and they are denoted by $\underset{1}{K}\dots \underset{5}{K}$.

For five independent tensors $\underset{\theta }{K}$ in [19], the invariants $\underset{\theta }{Z}$ were found, which are different from the invariants $\underset{\theta }{\tilde{Z}}$ in the present paper (see Remark 3.1, at the end). Compare e.g., $\underset{1}{\tilde{Z}}$ from the present paper and $\underset{1}{Z}$ from [19], where $\underset{1}{R}=\underset{1}{K}$.

Investigation of various kinds of mappings in the settings of generalized Riemannian spaces is an active research topic, numerous results were obtained in the recent years; see, for instance [20,21,22]. Very recently, conformal and concircular diffeomorphisms of generalized Riemannian spaces have been studied by M. Z Petrović, M. S. Stanković and P. Peška [23].

2. Equitorsion Conformal Transformation in Generalized Riemannian Space

Consider a special transformation of the objects in $G{R}_N$.

Definition 1.

Conformal transformation is that one under which the basic tensor is changed according to the law

$${\overline{g}}_{ij}(x)={\rho }^2(x)g_{ij}(x),(g_{ij}\ne g_{ji}),$$

(3)

where $\rho (x)=\rho (x^1,\dots ,x^N)$ is some differentiable function of coordinates in $G{R}_N$.

We see that g and $\overline{g}$ are considered in the common system of coordinates. The same is valid for the other geometric objects.

Furthermore, we have:

$$d{s}^2=g_{ij}d{x}^{i}d{x}^j,d{\overline{s}}^2={\overline{g}}_{ij}d{x}^{i}d{x}^j={\rho }^2{g}_{ij}d{x}^{i}d{x}^j,$$

$$d{\overline{s}}^2={\rho }^{2}d{s}^2\iff d\overline{s}/ds=\rho .$$

(4)

If the transformation (3) is effected, for the Christoffel symbols, it is obtained

$$\begin{array}{cc}{\overline{\Gamma }}_{i.jk}& =\frac{1}{2}({\overline{g}}_{ji,k}-{\overline{g}}_{jk,i}+{\overline{g}}_{ik,j})\hfill \\ & ={\rho }^2[\frac{{\rho }_{,k}}{\rho }{g}_{ji}-\frac{{\rho }_{,i}}{\rho }{g}_{jk}+\frac{{\rho }_{,j}}{\rho }{g}_{ik}+\frac{1}{2}(g_{ji,k}-g_{jk,i}+g_{ik,j})].\end{array}$$

Denoting

$${(ln\rho )}_{,i}=\frac{\partial (ln\rho )}{\partial x^i}=\frac{1}{\rho }{\rho }_{,i}={\rho }_i,$$

(5)

the previous equation gives

$${\overline{\Gamma }}_{i.jk}={\rho }^2({\Gamma }_{i.jk}+g_{ji}{\rho }_k-g_{jk}{\rho }_i+g_{ik}{\rho }_j).$$

(6)

For ${\overline{\Gamma }}_{jk}^i$, according to (1), we get

$${\overline{\Gamma }}_{jk}^i=\frac{1}{2}{\overline{g}}^{\underline{ip}}({\overline{g}}_{jp,k}-{\overline{g}}_{jk,p}+{\overline{g}}_{pk,j}).$$

(7)

Because the inverse matrix for $(g_{\underline{ij}})$ is the matrix $(g^{\underline{ij}})$, we get

$${\overline{g}}^{\underline{ij}}(x)={[\rho (x)]}^{-2}{g}^{\underline{ij}}(x)$$

(8)

and, based on (1), (6), (8),

$${\overline{\Gamma }}_{jk}^i={\Gamma }_{jk}^i+{\delta }_j^i{\rho }_k^{}+{\delta }_k^i{\rho }_j^{}-{\rho }_{}^i{g}_{\underline{jk}}^{}+{\xi }_{jk}^i,$$

(9)

where

$${\xi }_{jk}^i=g^{\underline{ip}}(g_{\underset{V}{jp}}{\rho }_k-g_{\underset{V}{jk}}{\rho }_p+g_{\underset{V}{pk}}{\rho }_j).$$

(10)

Denote

$${\rho }^i=g^{\underline{ip}}{\rho }_p\underset{(5)}{=}{g}^{\underline{ip}}(ln\rho ){}_{,p}.$$

(11)

From (9), it is obtained: for the symmetric part of the connection

$${\overline{\Gamma }}_{\underline{jk}}^i={\Gamma }_{\underline{jk}}^i+{\delta }_j^i{\rho }_k^{}+{\delta }_k^i{\rho }_j^{}-{\rho }_{}^i{g}_{\underline{jk}}^{},$$

(12)

and for the torsion tensor (double skewsymmetric part of the connection)

$${\overline{T}}_{jk}^i=2{\overline{\Gamma }}_{\underset{V}{jk}}^i=T_{jk}^i+2g^{\underline{ip}}(g_{\underset{V}{jp}}{\rho }_k-g_{\underset{V}{kp}}{\rho }_j+g_{\underset{V}{kj}}{\rho }_p)=T_{jk}^i+2{\xi }_{jk}^i.$$

(13)

Definition 2.

An equitorsion conformal transformation of the connection in $G{R}_N$ is that conformal transformation (3) on the base of which the torsion is not changed, i.e.,

$${\overline{T}}_{jk}^i={\overline{\Gamma }}_{jk}^i-{\overline{\Gamma }}_{kj}^i={\Gamma }_{jk}^i-{\Gamma }_{kj}^i=T_{jk}^i.$$

(14)

From (13), we conclude that

Theorem 1.

Necessary and sufficient condition for a conformal transformation of the connection to be equitorsion is

$${\xi }_{jk}^i=g^{\underline{ip}}(g_{\underset{V}{jp}}{\rho }_k-g_{\underset{V}{kp}}{\rho }_j+g_{\underset{V}{kj}}{\rho }_p)=0.$$

(15)

3. Curvature Tensors in Equitorsion Conformal and Concircular Transformation in Generalized Riemannian Space

3.1. The First Curvature Tensor

The 1^st from the cited curvature tensors in $G{R}_N$ is [12,13]

$$\underset{1}{R}{}_{jmn}^i={\Gamma }_{jm,n}^i-{\Gamma }_{jn,m}^i+{\Gamma }_{jm}^p{\Gamma }_{pn}^i-{\Gamma }_{jn}^p{\Gamma }_{pm}^i.$$

(16)

Based on (15), (9), we obtain

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

(17)

If by the transformation of the connection $\Gamma$ into $\overline{\Gamma }$ we write

$$a){\overline{\Gamma }}_{jk}^i={\Gamma }_{jk}^i+P_{jk}^i,b)P_{jk}^i={\delta }_j^i{\rho }_k^{}+{\delta }_k^i{\rho }_j^{}-{\rho }_{}^i{g}_{\underline{jk}}^{}=P_{kj}^i,$$

(18)

we can consider how e.g., some curvature tensors from the above mentioned independent ones are transformed.

With respect to (18), for $\underset{1}{R}$, one obtains

$$\begin{array}{c}\underset{1}{\overline{R}}{}_{jmn}^i={\overline{\Gamma }}_{jm,n}^i-{\overline{\Gamma }}_{jn,m}^i+{\overline{\Gamma }}_{jm}^p{\overline{\Gamma }}_{pn}^i-{\overline{\Gamma }}_{jn}^p{\overline{\Gamma }}_{pm}^i\\ =\underset{1}{R}{}_{jmn}^i+P_{jm\underset{1}{|}n}^i-P_{jn\underset{1}{|}m}^i+P_{jm}^p{P}_{pn}^i-P_{jn}^p{P}_{pm}^i+T_{mn}^p{P}_{jp}^i,\end{array}$$

(19)

and substituting P from (18b):

$$\begin{array}{c}\underset{1}{\overline{R}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i+{\delta }_j^i({\rho }_{m\underset{1}{|}n}-{\rho }_{n\underset{1}{|}m}+T_{mn}^p{\rho }_p)+{\delta }_m^i({\rho }_{j\underset{1}{|}n}+{\rho }_j{\rho }_n)\\ -{\delta }_n^i({\rho }_{j\underset{1}{|}m}-{\rho }_j{\rho }_m)+({\rho }_{\underset{1}{|}m}^i-{\rho }^i{\rho }_m)g_{\underline{jn}}-({\rho }_{\underset{1}{|}n}^i-{\rho }^i{\rho }_n)g_{\underline{jm}}\\ +{\rho }^p{\rho }_p({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}})+T_{mn}^i{\rho }_j-T_{j.mn}^{}{\rho }^i,\end{array}$$

(20)

where $\underset{1}{|}m$ denotes covariant derivative of the first kind on $x^m$. Because

$${\rho }_{m\underset{1}{|}n}-{\rho }_{n\underset{1}{|}m}=-T_{mn}^p{\rho }_p,$$

(21)

the 2nd addend on the right side in (20) is 0. Introducing the notation

$$\underset{1}{\rho }{}_{ij}={\rho }_{i\underset{1}{|}j}-{\rho }_i{\rho }_j+\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{ij}}={\rho }_{i\underset{1}{|}j}-{\rho }_i{\rho }_j+\frac{1}{2}{\rho }_r{\rho }^r{g}_{\underline{ij}},$$

(22)

we obtain

$$\underset{1}{\rho }{}_{mn}-\underset{1}{\rho }{}_{nm}\underset{(22)}{=}{\rho }_{m\underset{1}{|}n}-{\rho }_{n\underset{1}{|}m}=-T_{mn}^p{\rho }_p,$$

(23)

and, for $\overline{\underset{1}{R}}{}_{jmn}^i$,

$$\begin{array}{c}\overline{\underset{1}{R}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i+{\delta }_m^i(\underset{1}{\rho }{}_{jn}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{jn}})-{\delta }_n^i(\underset{1}{\rho }{}_{jm}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{jm}})\\ +g^{\underline{ip}}{g}_{\underline{jn}}({\rho }_{p\underset{1}{|}m}-{\rho }_p{\rho }_m)-g^{\underline{ip}}{g}_{\underline{jm}}({\rho }_{p\underset{1}{|}n}-{\rho }_p{\rho }_n)+{\rho }_p^{}{\rho }_{}^p({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})\\ +A_{jmn}^i\end{array}$$

is obtained, where

$$A_{jmn}^i=T_{mn}^i{\rho }_j-T_{j.mn}^{}{\rho }^i.$$

(24)

Furthermore,

$$\begin{array}{c}\overline{\underset{1}{R}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i+{\delta }_m^i(\underset{1}{\rho }{}_{jn}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{jn}})-{\delta }_n^i(\underset{1}{\rho }{}_{jm}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{jm}})\\ +g^{\underline{ip}}{g}_{\underline{jn}}(\underset{1}{\rho }{}_{pm}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{pm}})-g^{\underline{ip}}{g}_{\underline{jm}}(\underset{1}{\rho }{}_{pn}-\frac{1}{2}{g}^{\underline{rs}}{\rho }_r{\rho }_s{g}_{\underline{pn}})+{\rho }_p^{}{\rho }_{}^p({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})\\ +A_{jmn}^i,\end{array}$$

(25)

from where

$$\begin{array}{cc}{\overline{\underset{1}{R}}}^i{}_{jmn}& =\underset{1}{R}{}_{jmn}^i+{\delta }_m^i\underset{1}{\rho }{}_{jn}-{\delta }_n^i\underset{1}{\rho }{}_{jm}-{\delta }_m^i{g}_{\underline{jn}}{\rho }_p{\rho }^p+{\delta }_n^i{g}_{\underline{jm}}{\rho }_p{\rho }^p\\ & +{\rho }_m^i{g}_{\underline{jn}}-{\rho }_n^i{g}_{\underline{jm}}+{\rho }_p^{}{\rho }_{}^p({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})+A_{jmn}^i,\end{array}$$

and putting in order:

$${\overline{\underset{1}{R}}}^i{}_{jmn}=\underset{1}{R}{}_{jmn}^i+{\delta }_m^i\underset{1}{\rho }{}_{jn}-{\delta }_n^i\underset{1}{\rho }{}_{jm}+\underset{1}{\rho }{}_m^i{g}_{\underline{jn}}-\underset{1}{\rho }{}_n^i{g}_{\underline{jm}}+A_{jmn}^i,$$

(26)

where $A_{jmn}^i$ is given in (24). We are using the next definition from [2]

Definition 3.

If a conformal transformation in a Riemannian space $R_N:$

$$\overline{g}{}_{ij}={\rho }^{2}g{}_{ij},(g{}_{ij}=g{}_{ji})$$

transforms every geodesic circle into geodesic circle, the function $\rho (x)$ satisfies the partial differential equation

$${\rho }_{ij}=\mathsf{\Phi }(x)g_{ij}(x),(g_{ij}=g_{ji}),$$

(27)

where

$${\rho }_{ij}={\rho }_{i;j}-{\rho }_i{\rho }_j+\frac{1}{2}{\rho }_p{\rho }^p{g}_{ij},(g_{ij}=g_{ji}).$$

(28)

Such a transformation is calleda concircular transformationin $R_N$, andconcircular geometryis geometry that treats the concircular transformations and the spaces that allow such kinds of transformations.

In the $G{R}_N$, we consider transformations

$$\overline{g}{}_{ij}={\rho }^{2}g{}_{ij},(g{}_{ij}\ne g{}_{ji})$$

(29)

where, based on (22), ${\rho }_{i\underset{1}{|}j}\ne {\rho }_{j\underset{1}{|}i}$ in $G{R}_N$. Now, we take

$$\underset{1}{\rho }{}_{\underline{ij}}=\underset{1}{\mathsf{\Phi }}(x)g_{\underline{ij}}(x),(g_{ij}\ne g_{ji}),$$

(30)

and such a transformation we name a concircular transformation of the first kind in $G{R}_N$.

We have to find the function $\underset{1}{\mathsf{\Phi }}$. Substituting $\underset{1}{\rho }$ from (30) into (26), we get:

$$\overline{\underset{1}{R}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i+2\underset{1}{\mathsf{\Phi }}({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}})+A_{jmn}^i.$$

(31)

If we effect the contraction with $i=n,$ it follows that

$$\overline{\underset{1}{R}}{}_{jm}^{}=\underset{1}{R}{}_{jm}^{}+2\underset{1}{\mathsf{\Phi }}({\delta }_m^i{g}_{\underline{ji}}-{\delta }_i^i{g}_{\underline{jm}}),$$

where $\underset{1}{R}{}_{jm}^{}=\underset{1}{R}{}_{jmi}^i$, and so on, and we get:

$${\overline{\underset{1}{R}}}^{}{}_{jm}=\underset{1}{R}{}_{jm}^{}+2(1-N)\underset{1}{\mathsf{\Phi }}{g}_{\underline{jm}}+A_{jm}^{}.$$

By multiplying the corresponding sides of previous equation and the equation

$${\rho }^2{\overline{g}}^{\underline{jm}}=g^{\underline{jm}},$$

we obtain

$${\rho }^2\overline{\underset{1}{R}}=g^{\underline{jm}}\{\underset{1}{R}{}_{jm}^{}+2(1-N)\underset{1}{\mathsf{\Phi }}{g}_{\underline{jm}}+A_{jm}^{}\},$$

where ${\overline{\underset{1}{R}}}^{}{}_{jm}{\overline{g}}^{\underline{jm}}=\overline{\underset{1}{R}}{}_{}^{}$ and so on, while

$$A_{jm}{g}^{\underline{jm}}\underset{(24)}{=}{A}_{jmi}^i{g}^{\underline{jm}}=(T_{mi}^i{\rho }_j-T_{j.mi}^{}{\rho }^i)g^{\underline{jm}}=0,$$

and we get

$${\rho }^2\overline{\underset{1}{R}}=\underset{1}{R}+\underset{1}{\mathsf{\Phi }}[-2(N-1)N],$$

wherefrom it follows that

$$\underset{1}{\mathsf{\Phi }}(x)=-\frac{{\rho }^2\overline{\underset{1}{R}}-\underset{1}{R}}{2(N-1)N}.$$

(32)

Substituting $\underset{1}{\mathsf{\Phi }}$ into (31), we get

$$\overline{\underset{1}{R}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i-\frac{{\rho }^2\overline{\underset{1}{R}}-\underset{1}{R}}{(N-1)N}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})+A_{jmn}^i$$

(33)

and from here

$$\begin{array}{c}\overline{\underset{1}{R}}{}_{jmn}^i+\frac{\underset{1}{\overline{R}}{}_{}^{}({\rho }^2{\delta }_m^i{g}_{jn}^{}-{\rho }^2{\delta }_n^i{g}_{jm}^{})}{(N-1)N}\\ =\underset{1}{R}{}_{jmn}^i+\frac{\underset{1}{R}{}_{}^{}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})}{(N-1)N}+A_{jmn}^i.\end{array}$$

(34)

Taking into consideration that

$${\rho }_i=\frac{1}{2N}[{(ln\overline{g})}_{,i}-{(lng)}_{,i}]=\frac{1}{2N}({\overline{g}}_i-g_i),g=det(g_{ij}),$$

(35)

with respect to (24) and (35)

$$\begin{array}{c}{A}_{jmn}^i=T_{mn}^i{\rho }_j-T_{j.mn}^{}{\rho }^i=T_{mn}^i{\rho }_j-T_{j.mn}^{}{g}^{\underline{ip}}{\rho }_p\underset{(35)}{=}\frac{1}{2N}[T_{mn}^i({\overline{g}}_j-g_j)-T_{j.mn}^{}{g}^{\underline{ip}}({\overline{g}}_p-g_p)]\\ =\frac{1}{2N}[({\overline{T}}_{mn}^i{\overline{g}}_j-{\overline{T}}_{j.mn}^{}{\overline{g}}^{\underline{ip}}{\overline{g}}_p)-(T_{mn}^i{g}_j-T_{j.mn}^{}{g}^{\underline{ip}}{g}_p)],\end{array}$$

(36)

where $T_{mn}^i=\overline{T}{}_{mn}^i$ (for the first addend) and $g^{\underline{ip}}{g}_{\underline{qj}}={\overline{g}}^{\underline{ip}}{\overline{g}}_{\underline{qj}}$ (for the third addend). By substituting from (36) into (34) and because of

$$g_m^i={\overline{g}}_m^i,{\rho }^2{g}_{jm}={\overline{g}}_{jm},\overline{\delta }{}_m^i={\delta }_m^i,$$

(37)

we obtain

$$\begin{array}{c}\overline{\underset{1}{R}}{}_{jmn}^i+\frac{\underset{1}{\overline{R}}{}_{}^{}({\overline{\delta }}_m^i{\overline{g}}_{\underline{jn}}^{}-{\overline{\delta }}_n^i{\overline{g}}_{\underline{jm}}^{})}{(N-1)N}+\frac{1}{2N}({\overline{T}}_{j.mn}^{}{\overline{g}}^{\underline{ip}}{\overline{g}}_p-{\overline{T}}_{mn}^i{\overline{g}}_j^{}),\\ =\underset{1}{R}{}_{jmn}^i+\frac{\underset{1}{\overline{R}}{}_{}^{}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})}{(N-1)N}+\frac{1}{2N}(T_{j.mn}^{}{g}^{\underline{ip}}{g}_p-T_{mn}^i{g}_j^{}).\end{array}$$

(38)

In that manner, we conclude that the following theorem is valid:

Theorem 2.

The tensor

$$\begin{array}{c}\underset{1}{\tilde{Z}}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i+\frac{\underset{1}{R}{}_{}^{}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})}{(N-1)N}\\ +\frac{1}{2N}(T_{j.mn}^{}{g}^{\underline{ip}}{g}_p-T_{mn}^i{g}_j^{})\end{array}$$

(39)

is an invariant in the space $G{R}_N$, by an equitorsion concircular transformation i.e., according to (38):

$$\underset{1}{\overline{\tilde{Z}}}{}_{jmn}^i=\underset{1}{\tilde{Z}}{}_{jmn}^i,$$

(40)

where e.g., $g_j={(lng)}_{,j}=\frac{\partial (lng)}{\partial x^j}$ and $\underset{1}{\tilde{Z}}{}_{jmn}^i$ is given by (39).

The tensor $\underset{1}{\tilde{Z}}{}_{jmn}^i$ is an equitorsion concircular tensor of the first kind in $G{R}_N$.

3.2. The Second Curvature Tensor

The tensor $\underset{2}{R}$ in $G{R}_N$ is [12,17]

$$\underset{2}{R}{}_{jmn}^i={\Gamma }_{mj,n}^i-{\Gamma }_{nj,m}^i+{\Gamma }_{mj}^p{\Gamma }_{np}^i-{\Gamma }_{nj}^p{\Gamma }_{mp}^i,$$

(41)

and, for $\underset{2}{\overline{R}}{}_{jmn}^i$, by virtue of (18), it follows that

$$\underset{2}{\overline{R}}{}_{jmn}^i={\underset{2}{R}}_{jmn}^i+P_{mj\underset{2}{|}n}^i-P_{nj\underset{2}{|}m}^i+P_{mj}^p{P}_{mp}^i-P_{nj}^p{P}_{mp}^i-T_{mn}^p{P}_{pj}^i.$$

(42)

Substituting from (18) into the previous equation and arranging, one obtains

$$\begin{array}{c}\underset{2}{\overline{R}}{}_{jmn}^i=\underset{2}{R}{}_{mj,n}^i+{\delta }_j^i({\rho }_{m\underset{2}{|}n}^{}-{\rho }_{n\underset{2}{|}m}^{}-T_{mn}^p{\rho }_p^{})+{\delta }_m^i({\rho }_{j\underset{2}{|}n}^{}-{\rho }_i^{}{\rho }_n^{})-{\delta }_n^i({\rho }_{j\underset{2}{|}m}^{}-{\rho }_i^{}{\rho }_m^{})\\ +({\rho }_{\underset{2}{|}m}^i-{\rho }_{}^i{\rho }_m^{})g_{\underline{jn}}-({\rho }_{\underset{2}{|}n}^i-{\rho }_{}^i{\rho }_n^{})g_{\underline{jm}}+{\rho }_p^{}{\rho }_{}^p({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}})-T_{mn}^i{\rho }_j^{}+T_{j.mn}^{}{\rho }_{}^i.\end{array}$$

(43)

The term in the $1st$ bracket on the right side is 0 because of

$${\rho }_{m\underset{2}{|}n}^{}-{\rho }_{n\underset{2}{|}m}^{}=T_{mn}^p{\rho }_p^{}.$$

(44)

If we introduce the denotation

$$\underset{2}{\rho }{}_{ij}^{}={\rho }_{i\underset{2}{|}j}^{}-{\rho }_i^{}{\rho }_j^{}+\frac{1}{2}{\rho }_r^{}{\rho }_{}^r{g}_{\underline{ij}},$$

(45)

we have

$$\underset{2}{\rho }{}_{mn}^{}-\underset{2}{\rho }{}_{nm}^{}=\rho {}_{m\underset{2}{|}n}^{}-\rho {}_{n\underset{2}{|}m}^{}=T_{mn}^p{\rho }_p^{}$$

and, for $\overline{\underset{2}{R}}{}_{jmn}^i$ from (43)–(45), it follows that

$$\overline{\underset{2}{R}}{}_{jmn}^i=\underset{2}{R}{}_{jmn}^i+{\delta }_m^i\underset{2}{\rho }{}_{jn}-{\delta }_n^i\underset{2}{\rho }{}_{jm}+\underset{2}{\rho }{}_m^i{g}_{\underline{jn}}-\underset{2}{\rho }{}_n^i{g}_{\underline{jm}}-A_{jmn}^i,$$

(46)

where $A_{jmn}^i$ is given (24). Furthermore, we use the concircular transformation for $\underset{2}{R}$

$$\underset{2}{\rho }{}_{\underline{ij}}^{}=\underset{2}{\mathsf{\Phi }}(x)g_{\underline{ij}}(x).$$

(47)

By substitution of $\underset{2}{\rho }{}_{ij}$ into (46), by procedure as for $\underset{1}{R}$, we obtain

$$\underset{2}{\mathsf{\Phi }}=-\frac{{\rho }^2\underset{2}{\overline{R}}-\underset{2}{R}}{2(N-1)N}$$

(48)

and at the end:

$$\overline{\underset{2}{R}}{}_{jmn}^i=\underset{2}{R}{}_{jmn}^i-\frac{{\rho }^2\underset{2}{\overline{R}}-\underset{2}{R}}{(N-1)N}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})-A_{jmn}^i,$$

(49)

where $A_{jmn}^i$ is given in (24).

Thus, we conclude that the next theorem is valid.

Theorem 3.

The tensor

$$\begin{array}{cc}{\widetilde{\underset{2}{Z}}}^{}{i}_{jmn}& =\underset{2}{R}{}_{jmn}^i+\frac{\underset{2}{R}({\delta }_m^i{g}_{jn}^{}-{\delta }_n^i{g}_{jm}^{})}{(N-1)N}\\ & -\frac{1}{2N}(T_{j.mn}^{}{g}^{\underline{ip}}{g}_p-T_{mn}^i{g}_j^{})\end{array}$$

(50)

is an invariant in $G{R}_N$ with respect to an equitorsion concircular transformation, i.e., in force is

$$\overline{\widetilde{\underset{2}{Z}}}{}_{jmn}^i=\widetilde{\underset{2}{Z}}{}_{jmn}^i.$$

(51)

The tensor $\widetilde{\underset{2}{Z}}$ isan equitorsion concircular tensor of the $2nd$ kind at $G{R}_N$and e.g., $g_j={(lng)}_{=}\frac{\partial (lng)}{\partial x^j}$.

3.3. The Third Curvature Tensor

The tensor $\underset{3}{R}$ in $G{R}_N$ [12,14,17] is

$$\underset{3}{R}{}_{jmn}^i={\Gamma }_{jm,n}^i-{\Gamma }_{nj,m}^i+{\Gamma }_{jm}^p{\Gamma }_{np}^i-{\Gamma }_{nj}^p{\Gamma }_{pm}^i+{\Gamma }_{nm}^p{T}_{pj}^i,$$

(52)

where $T_{pj}^i$ is torsion tensor in local coordinates. For $\underset{3}{R}{}_{jmn}^i$ on the base of (18), it is obtained

$$\underset{3}{\overline{R}}{}_{jmn}^i=\underset{3}{R}{}_{jmn}^i+P_{jm\underset{2}{|}n}^i-P_{nj\underset{1}{|}m}^i+P_{np}^i{P}_{jm}^p-P_{pm}^i{P}_{nj}^p+T_{pj}^i{P}_{nm}^p,$$

(53)

where we take into consideration that $P_{jp}^i$ is symmetric, with respect to (18).

By substituting from (18) into the previous equation and arranging, one obtains

$$\begin{array}{c}\underset{3}{\overline{R}}{}_{jmn}^i=\underset{3}{R}{}_{jmn}^i+{\delta }_m^i(\underset{2}{\rho }{}_{jn}^{}-\frac{1}{2}{\rho }_p{\rho }^p{g}_{\underline{jn}})-{\delta }_n^i(\underset{1}{\rho }{}_{jm}^{}-\frac{1}{2}{\rho }_p{\rho }^p-g_{\underline{jm}})\\ +g_{}^{\underline{ip}}{g}_{\underline{jn}}^{}(\underset{1}{\rho }{}_{pm}^{}-\frac{1}{2}{\rho }_r^{}{\rho }_{}^r{g}_{\underline{pm}}^{})-g_{}^{\underline{ip}}{g}_{\underline{jm}}^{}(\underset{2}{\rho }{}_{pn}^{}-\frac{1}{2}{\rho }_r^{}{\rho }_{}^r{g}_{\underline{pm}}^{})\\ +{\rho }_p^{}{\rho }_{}^p{\delta }_m^i{g}_{\underline{jn}}-{\rho }_p^{}{\rho }_{}^p{\delta }_n^i{g}_{\underline{jm}}+D_{jmn}^i,\end{array}$$

(54)

where

$$D_{jmn}^i=T_{jm}^i{\rho }_n^{}+T_{nj}^i{\rho }_m^{}+g^{\underline{ps}}{g}_{\underline{mn}}{T}_{jp}^i{\rho }_s^{}.$$

(55)

From (55), it is obtained that

$$\underset{3}{\overline{R}}{}_{jmn}^i=\underset{3}{R}{}_{jmn}^i+{\delta }_m^i\underset{2}{\rho }{}_{jn}^{}-{\delta }_n^i\underset{1}{\rho }{}_{jm}^{}+\underset{1}{\rho }{}_m^i{g}_{\underline{jn}}-\underset{2}{\rho }{}_n^i{g}_{\underline{jm}}+D_{jmn}^i.$$

(56)

Consider, further, the concircular transformation for the tensor $\underset{3}{R}{}_{jmn}^i$ in the following manner.

Taking

$$\underset{\theta }{\rho }{}_{\underline{ij}}=\underset{3}{\mathsf{\Phi }}(x)g_{\underline{ij}}(x),\theta =1,2,$$

(57)

we obtain from (56)

$$\underset{3}{\overline{R}}{}_{jmn}^i=\underset{3}{R}{}_{jmn}^i+2\underset{3}{\mathsf{\Phi }}{\delta }_m^i{g}_{\underline{jn}}-({\delta }_n^i{g}_{\underline{jm}})+D_{jmn}^i.$$

(58)

Putting $i=n,$ we get

$$\underset{3}{\overline{R}}{}_{jm}^{}=\underset{3}{R}{}_{jm}^{}+2\underset{3}{\mathsf{\Phi }}{\delta }_m^i{g}_{\underline{ji}}-({\delta }_i^i{g}_{\underline{jm}})+D_{jm}^{},$$

(59)

and contracting with ${\rho }^2\overline{g}{}^{jm}=g^{\underline{jm}}$ on the left and the right sides correspondingly in (59), we get

$${\rho }^2\underset{3}{\overline{R}}{}_{}^{}=\underset{3}{R}{}_{}^{}+2\underset{3}{\mathsf{\Phi }}(1-N)N$$

(60)

because

$$\begin{array}{ccc}D& =D_{jm}{g}^{\underline{jm}}& =D_{jmi}^i{g}^{\underline{jm}}\underset{(55)}{=}(T_{jm}^i{\rho }_i^{}+T_{ij}^i{\rho }_m^{}+g^{\underline{ps}}{g}_{\underline{mi}}{T}_{jp}^i{\rho }_s^{})g^{\underline{jm}}\\ & & =T_{jm}^i{\rho }_i^{}{g}^{\underline{jm}}+0+g^{\underline{ps}}{\delta }_i^j{T}_{jp}^i{\rho }_s^{}=T_{jm}^i{\rho }_i^{}{g}^{\underline{jm}}\\ & & =T_{i.jm}^{}{\rho }_{}^i{g}^{\underline{jm}}=-T_{j.im}^{}{\rho }_{}^i{g}^{\underline{jm}}=-T_{im}^m{\rho }_{}^i=0.\end{array}$$

(61)

By the further procedure as in the case of $\underset{1}{R}$, we obtain

$$\underset{3}{\mathsf{\Phi }}(x)=-\frac{{\rho }^2\underset{3}{\overline{R}}-\underset{3}{R}}{(N-1)N}.$$

(62)

Consider, further, the tensor $D_{jmn}^i$. By virtue of (35), one gets

$$D_{jmn}^i=\frac{1}{2N}[T_{jm}^i({\overline{g}}_n-g_n)+T_{nj}^i({\overline{g}}_m-g_m)+g^{\underline{ps}}{g}_{\underline{mn}}{T}_{jp}^i({\overline{g}}_s-g_s)],$$

(63)

where the equitorsion is taken into consideration.

Substituting from (62), (63) into (58), it follows that

$$\begin{array}{c}\underset{3}{\overline{R}}{}_{jmn}^i=\underset{3}{R}{}_{jmn}^i-\frac{{\rho }^2\underset{3}{\overline{R}}-\underset{3}{R}}{(N-1)N}({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}})\\ +\frac{1}{2N}[T_{jm}^i({\overline{g}}_n-g_n)+T_{nj}^i({\overline{g}}_m-g_m)+g^{\underline{ps}}{g}_{\underline{mn}}{T}_{jp}^i({\overline{g}}_s-g_s)].\end{array}$$

(64)

from where we conclude that the next theorem is valid.

Theorem 4.

The tensor

$$\begin{array}{cc}{\widetilde{\underset{3}{Z}}}^{}{i}_{jmn}& =\underset{3}{R}{}_{jmn}^i+\frac{\underset{3}{R}({\delta }_m^i{g}_{jn}^{}-{\delta }_n^i{g}_{jm}^{})}{(N-1)N}\\ & -\frac{1}{2N}(T_{jm}^i{g}_n^{}+T_{nj}^i{g}_m+g_{}^{\underline{ps}}{g}_{\underline{mn}}{T}_{jp}^i{g}_s)\end{array}$$

(65)

is an invariant in $G{R}_N$ with respect to an equitorsion concircular transformation, i.e., it is

$$\underset{3}{\overline{\tilde{Z}}}{}_{jmn}^i=\widetilde{\underset{3}{Z}}{}_{jmn}^i.$$

(66)

The tensor $\widetilde{\underset{3}{Z}}$ isan equitorsion concircular tensor of the 3rd kind at $G{R}_N$.

3.4. The Fourth Curvature Tensor

For the tensor $\underset{4}{R}$ in $G{R}_N$, we have [13,14,17]

$$\underset{4}{R}{}_{jmn}^i={\Gamma }_{jm,n}^i-{\Gamma }_{nj,m}^i+{\Gamma }_{jm}^p{\Gamma }_{np}^i-{\Gamma }_{nj}^p{\Gamma }_{pm}^i+{\Gamma }_{mn}^p{T}_{pj}^i,$$

(67)

where $T_{pj}^i$ is torsion tensor in local coordinates. For $\underset{4}{R}{}_{jmn}^i$ on the base of (18), it is obtained

$$\underset{4}{\overline{R}}{}_{jmn}^i={\underset{4}{R}}_{jmn}^i+P_{jm\underset{2}{|}n}^i-P_{nj\underset{1}{|}m}^i+P_{np}^i{P}_{jm}^p-P_{pm}^i{P}_{nj}^p+T_{pj}^i{P}_{mn}^p.$$

(68)

From (53), (68), it follows that

$$\underset{4}{\overline{R}}{}_{jmn}^i-\underset{3}{\overline{R}}{}_{jmn}^i=\underset{4}{R}{}_{jmn}^i-\underset{3}{R}{}_{jmn}^i+2P{}_{\underset{V}{mn}}^p{T}_{pj}^i=\underset{4}{R}{}_{jmn}^i-\underset{3}{R}{}_{jmn}^i$$

because $P{}_{\underset{V}{mn}}^p=0.$ Thus, we have

$$\begin{array}{c}\underset{4}{\overline{R}}{}_{jmn}^i-\underset{4}{R}{}_{jmn}^i=\underset{3}{\overline{R}}{}_{jmn}^i-\underset{3}{R}{}_{jmn}^i\\ \underset{(56)}{=}{\delta }_m^i\underset{2}{\rho }{}_{jn}^{}-{\delta }_n^i\underset{1}{\rho }{}_{jm}^{}+\underset{1}{\rho }{}_m^i{g}_{\underline{jn}}-\underset{2}{\rho }{}_n^i{g}_{\underline{jm}}+D_{jmn}^i,\end{array}$$

(69)

where $D_{jmn}^i$ is given in (55). For the concircular transformation for the tensor $\underset{4}{R}{}_{jmn}^i$, we put

$$\underset{\theta }{\rho }{}_{\underline{ij}}=\underset{4}{\mathsf{\Phi }}(x)g_{\underline{ij}}(x),\theta =1,2,$$

and, by the same procedure as in the previous case, the next theorem is obtained.

Theorem 5.

The tensor

$$\begin{array}{cc}\underset{4}{\tilde{Z}}{}_{jmn}^i& =\underset{4}{R}{}_{jmn}^i+\frac{\underset{4}{R}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})}{(N-1)N}\\ & -\frac{1}{2N}(T_{jm}^i{g}_n+T_{nj}^i{g}_m+g^{\underline{ps}}{g}_{\underline{mn}}{T}_{jp}^i{g}_s)\end{array}$$

(70)

is an invariant in $G{R}_N$ with respect to an equitorsion concircular transformation, i.e., in force is

$$\underset{4}{\overline{\tilde{Z}}}{}_{jmn}^i=\widetilde{\underset{4}{Z}}{}_{jmn}^i.$$

(71)

The tensor $\widetilde{\underset{4}{Z}}$ isan equitorsion concircular tensor of the 4th kind at $G{R}_N$.

3.5. The Fifth Curvature Tensor

Finally, consider the 5th curvature tensor $\underset{5}{R}{}_{jmn}^i$ in $G{R}_N$ (in [12] $\underset{5}{R}$ is denoted with $\underset{2}{\tilde{R}})$. We have according to [12,17]

$$\begin{array}{cc}\underset{5}{R}{}_{jmn}^i& =\frac{1}{2}({\Gamma }_{jm,n}^i+{\Gamma }_{mj,n}^i-{\Gamma }_{jn,m}^i-{\Gamma }_{nj,m}^i\\ & +{\Gamma }_{jm}^p{\Gamma }_{pn}^i+{\Gamma }_{mj}^p{\Gamma }_{np}^i-{\Gamma }_{jn}^p{\Gamma }_{mp}^i-{\Gamma }_{nj}^p{\Gamma }_{pm}^i),\end{array}$$

(72)

which can be written in the form [17]:

$$\begin{array}{cc}\underset{5}{\overline{R}}{}_{jmn}^i=& \underset{5}{R}{}_{jmn}^i+\frac{1}{2}(P_{jm\underset{3}{|}n}^i-P_{jn\underset{4}{|}m}^i+P_{mj\underset{4}{|}n}^i-P_{nj\underset{3}{|}m}^i\\ & +P_{jm}^p{P}_{pn}^i-P_{jn}^p{P}_{mp}^i+P_{mj}^p{P}_{np}^i-P_{nj}^p{P}_{pm}^i),\end{array}$$

(73)

where $P_{jk}^i$ is given in (18). With substitution of P from (18) into (73), one obtains

$$\begin{array}{c}\underset{5}{\overline{R}}{}_{jmn}^i=\underset{5}{R}{}_{jmn}^i+\frac{1}{2}[{\delta }_j^i({\rho }_{m\underset{2}{|}n}^{}-{\rho }_{n\underset{1}{|}m}^{}-{\rho }_{m\underset{1}{|}n}^{}-{\rho }_{n\underset{2}{|}m}^{})\\ +{\delta }_m^i({\rho }_{j\underset{2}{|}n}^{}+{\rho }_{j\underset{1}{|}n}^{}-2{\rho }_j^{}{\rho }_n^{}+{\rho }_p{\rho }^p{g}_{\underline{jn}})\\ -{\delta }_n^i({\rho }_{j\underset{1}{|}m}^{}+{\rho }_{j\underset{2}{|}m}^{}-2{\rho }_j^{}{\rho }_m^{}+{\rho }_p{\rho }^p{g}_{\underline{jm}})\\ +g_{\underline{jn}}({\rho }_{\underset{1}{|}m}^i+{\rho }_{\underset{2}{|}m}^i-2{\rho }^i{\rho }_m^{})-g_{\underline{jm}}({\rho }_{\underset{1}{|}n}^i+{\rho }_{\underset{2}{|}n}^i-2{\rho }^i{\rho }_n^{})].\end{array}$$

(74)

Using (23) and (44) and introducing the denotation

$$\underset{5}{\rho }{}_{ij}^{}=\frac{1}{2}({\rho }_{i\underset{1}{|}j}^{}+{\rho }_{i\underset{2}{|}j}^{}-2{\rho }_i^{}{\rho }_j^{}+{\rho }_p^{}{\rho }_{}^p{g}_{\underline{ij}})=\frac{1}{2}(\underset{1}{\rho }{}_{ij}^{}+\underset{2}{\rho }{}_{ij}^{})=\underset{5}{\rho }{}_{ji}^{},$$

(75)

Equation (74) obtains the form

$$\begin{array}{c}\underset{5}{\overline{R}}{}_{jmn}^i=\underset{5}{R}{}_{jmn}^i+{\delta }_m^i\underset{5}{\rho }{}_{jn}^{}-{\delta }_n^i\underset{5}{\rho }{}_{jm}^{}\\ +\underset{5}{\rho }{}_m^i{g}_{\underline{jn}}-\underset{5}{\rho }{}_n^i{g}_{\underline{jm}}+\rho {}_p^{}\rho {}_{}^p({\delta }_n^i{g}_{\underline{jm}}-{\delta }_m^i{g}_{\underline{jm}}).\end{array}$$

(76)

Let us apply a concircular transformation for the tensor $\underset{5}{R}{}_{jmn}^i$. By virtue of (75), we put

$$\underset{5}{\rho }{}_{ij}=\underset{5}{\mathsf{\Phi }}(x)g_{\underline{ij}}(x)=\underset{5}{\rho }{}_{ji},$$

(77)

into (76) and we get

$$\underset{5}{\overline{R}}{}_{jmn}^i=\underset{5}{R}{}_{jmn}^i+2\underset{5}{\mathsf{\Phi }}({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}})-\rho {}_p^{}\rho {}_{}^p({\delta }_m^i{g}_{\underline{jn}}-{\delta }_n^i{g}_{\underline{jm}}).$$

(78)

Contracting by $i=n$, we obtain

$$\underset{5}{\overline{R}}{}_{jm}^{}=\underset{5}{R}{}_{jm}^{}+(1-N)g_{\underline{jm}}(2\underset{5}{\mathsf{\Phi }}-\rho {}_p^{}\rho {}_{}^p).$$

Multiplying this equation with ${\rho }^2\overline{g}{}^{jm}=g{}^{\underline{jm}}$, it follows that

$$2\underset{5}{\mathsf{\Phi }}=\frac{{\rho }^2\underset{5}{\overline{R}}-\underset{5}{R}}{(1-N)N}+\rho {}_p^{}\rho {}_{}^p$$

and substituting this value into (78), one gets that the following theorem is valid.

Theorem 6.

The tensor

$$\widetilde{\underset{5}{Z}}{}_{jmn}^i=\underset{5}{R}{}_{jmn}^i+\frac{\underset{5}{R}({\delta }_m^i{g}_{\underline{jn}}^{}-{\delta }_n^i{g}_{\underline{jm}}^{})}{(N-1)N}$$

(79)

is an invariant in $G{R}_N$ with respect to an equitorsion concircular transformation, i.e., in force is

$$\underset{5}{\overline{\tilde{Z}}}{}_{jmn}^i=\widetilde{\underset{5}{Z}}{}_{jmn}^i.$$

(80)

The tensor $\underset{5}{\tilde{Z}}$ isan equitorsion concircular tensor of the 5th kind at $G{R}_N$.

Remark 1.

In [19], is $\underset{1}{K}{}_{jmn}^i=\underset{1}{R}{}_{jmn}^i,$ $\underset{3}{K}{}_{jmn}^i=\underset{3}{R}{}_{jm}^i,$ while $\underset{\theta }{R}{}_{jm}^i\notin \{\underset{2}{K}{}_{jmn}^i,\underset{4}{K}{}_{jmn}^i,$ $\underset{5}{K}{}_{jmn}^i\},\theta =2,4,5.$ However, because of different procedures, it is $\underset{\theta }{\tilde{Z}}{}_{jmn}^i\notin \{\underset{1}{Z}{}_{jmn}^i,\dots ,$ $\underset{5}{Z}{}_{jmn}^i\},\theta =1,\dots ,5,$ where $\underset{\theta }{Z}{}_{jmn}^i$ are from [19]. Thus, $\underset{\theta }{\tilde{Z}}{}_{jmn}^i$ arenew invariantsof the considered transformations.

Remark 2.

In the case of $R_N(g_{ij}=g_{ji},T_{jk}^i=0)$, each of the obtained tensors $\underset{\theta }{\tilde{Z}}{}_{jmn}^i$ reduces to a known concircular tensor [2] $Z_{jmn}^i=R_{jmn}^i+\frac{R({\delta }_m^i{g}_{jn}-{\delta }_n^i{g}_{jm})}{(N-1)N}$.

4. Conclusions

Conformal equitorsion concircular transformations are investigated and corresponding invariants-5 concircular tensors of concircular transformations are found.