Associated Derived Invariants Of Geometric Mappings

TL;DR

This paper derives general invariants for geometric mappings in symmetric affine connection spaces, extending previous invariants and relating them to the Thomas projective parameter and Weyl tensor.

Contribution

It introduces new general invariants for geometric mappings, expanding the theoretical framework beyond existing basic invariants.

Findings

Derived new invariants related to geometric mappings

Connected invariants with Thomas projective parameter

Linked invariants to Weyl projective tensor

Abstract

General invariants of a geometric mapping of a symmetric affine connection space are obtained in this paper. These invariants are generalizations of the previous obtained basic invariants (see [16]). Moreover, these invariants are related with the Thomas projective parameter and the Weyl projective tensor.

Equations

$$L^{i}_{\underline{jk}}=\frac{1}{2}\big{(}L^{i}_{jk}+L^{i}_{kj}\big{)},$$

$$L_{\underline{j^{\prime }{k}^{\prime }}}^{i^{\prime }}=x_i^{i^{\prime }}{x}_{j^{\prime }}^j{x}_{k^{\prime }}^k{L}_{\underline{jk}}^i+x_i^{i^{\prime }}{x}_{j^{\prime }{k}^{\prime }}^i,$$

$$a_{j|k}^i=a_{j,k}^i+L_{\underline{\alpha k}}^i{a}_j^{\alpha }-L_{\underline{jk}}^{\alpha }{a}_{\alpha }^i,$$

$$R_{jmn}^i=L_{\underline{jm},n}^i-L_{\underline{jn},m}^i+L_{\underline{jm}}^{\alpha }{L}_{\underline{\alpha n}}^i-L_{\underline{jn}}^{\alpha }{L}_{\underline{\alpha m}}^i.$$

$$T^{i}_{jk}=L^{i}_{\underline{jk}}-\frac{1}{N+1}\big{(}\delta^{i}_{k}L^{\alpha}_{\underline{j\alpha}}+\delta^{i}_{j}L^{\alpha}_{\underline{k\alpha}}\big{)}.$$

$$W_{jmn}^i=R_{jmn}^i+\frac{1}{N+1}{\delta }_j^i{R}_{[mn]}+\frac{N}{N^2-1}{\delta }_{[m}^i{R}_{jn]}+\frac{1}{N^2-1}{\delta }_{[m}^i{R}_{n]j}.$$

$$W^{i}_{jmn}=R^{i}_{jmn}+\frac{1}{N-1}\big{(}\delta^{i}_{m}R_{jn}-\delta^{i}_{n}R_{jm}\big{)}.$$

$$P_{jk}^i={\overline{L}}_{jk}^i-L_{jk}^i=\overline{\omega }{}_{jk}^i-{\omega }_{jk}^i+\overline{\tau }{}_{jk}^i-{\tau }_{jk}^i,$$

$${\overline{L}}_{\underline{jk}}^i=L_{\underline{jk}}^i+{\overline{\omega }}_{jk}^i-{\omega }_{jk}^i.$$

$$\begin{array}{r}\tilde{\mathcal{T}}{}_{jk}^i=L_{\underline{jk}}^i-{\omega }_{jk}^i,\end{array}$$

$$\begin{array}{r}\tilde{\mathcal{W}}{}_{jmn}^i=R_{jmn}^i-\omega {}_{jm|n}^i+\omega {}_{jn|m}^i+\omega {}_{jm}^{\alpha }\omega {}_{\alpha n}^i-\omega {}_{jn}^{\alpha }\omega {}_{\alpha m}^i,\end{array}$$

$$\omega^{i}_{jk}=s_{(1)}\big{(}\delta^{i}_{j}\rho_{k}+\delta^{i}_{k}\rho_{j}\big{)}+s_{(2)}\big{(}F^{i}_{j}\sigma_{k}+F^{i}_{k}\sigma_{j}\big{)}+s_{(3)}\sigma_{jk}\varphi^{i},$$

$${\omega }_{jm}^{\alpha }{\omega }_{\alpha n}^i$$

$$\displaystyle+\delta^{i}_{n}\Big{(}2\big{(}s_{(1)}\big{)}^{2}\rho_{j}\rho_{m}+s_{(1)}s_{(2)}\big{(}F^{\alpha}_{m}\rho_{\alpha}\sigma_{j}+F^{\alpha}_{j}\rho_{\alpha}\sigma_{m}\big{)}+s_{(1)}s_{(3)}\sigma_{jm}\rho_{\alpha}\varphi^{\alpha}\Big{)}$$

$$\displaystyle+\big{(}s_{(2)}\big{)}^{2}\Big{(}F^{i}_{n}\big{(}F^{\alpha}_{m}\sigma_{j}+F^{\alpha}_{j}\sigma_{m}\big{)}\sigma_{\alpha}+F^{i}_{\alpha}\big{(}F^{\alpha}_{m}\sigma_{j}+F^{\alpha}_{j}\sigma_{m}\big{)}\sigma_{n}\Big{)}+\big{(}s_{(3)}\big{)}^{2}\sigma_{jm}\sigma_{\alpha n}\varphi^{\alpha}\varphi^{i}$$

$$\displaystyle+s_{(1)}s_{(2)}\Big{(}F^{i}_{n}\big{(}\rho_{j}\sigma_{m}+\rho_{m}\sigma_{j}\big{)}+F^{i}_{m}\big{(}\rho_{j}\sigma_{n}+\rho_{n}\sigma_{j}\big{)}+F^{i}_{j}\big{(}\rho_{m}\sigma_{n}+\rho_{n}\sigma_{m}\big{)}\Big{)}$$

$$\displaystyle+s_{(1)}s_{(3)}\big{(}\sigma_{mn}\rho_{j}+\sigma_{jn}\rho_{m}+\sigma_{jm}\rho_{n}\big{)}\varphi^{i}$$

$$\displaystyle+s_{(2)}s_{(3)}\Big{(}\big{(}F^{\alpha}_{m}\sigma_{j}+F^{\alpha}_{j}\sigma_{m}\big{)}\sigma_{\alpha n}\varphi^{i}+\big{(}F^{i}_{n}\sigma_{\alpha}+F^{i}_{\alpha}\sigma_{n}\big{)}\sigma_{jm}\varphi^{\alpha}\Big{)}$$

$$\displaystyle\begin{aligned} \widetilde{\mathcal{T}}{}^{(s).i}_{jk}=L^{i}_{\underline{jk}}-s_{(1)}\big{(}\delta^{i}_{j}\rho_{k}+\delta^{i}_{k}\rho_{j}\big{)}-s_{(2)}\big{(}F^{i}_{j}\sigma_{k}+F^{i}_{k}\sigma_{j}\big{)}-s_{(3)}\sigma_{jk}\varphi^{i},\end{aligned}$$

$$\begin{array}{rl}\tilde{\mathcal{W}}{}_{jmn}^{(s).i}& =R_{jmn}^i-{\delta }_j^i{\zeta }_{[mn]}^{(s)}-{\delta }_m^i{\zeta }_{jn}^{(s)}+{\delta }_n^i{\zeta }_{jm}^{(s)}+\tilde{\mathcal{D}}{}_{jmn}^{(s_2).(s_3).i}-\tilde{\mathcal{D}}{}_{jnm}^{(s_2).(s_3).i},\end{array}$$

$$\displaystyle\begin{aligned} \zeta^{(s)}_{ij}&=s_{(1)}\rho_{i|j}+\big{(}s_{(1)}\big{)}^{2}\rho_{i}\rho_{j}+s_{(1)}s_{(2)}\big{(}F^{\alpha}_{i}\sigma_{j}+F^{\alpha}_{j}\sigma_{i}\big{)}\rho_{\alpha}+s_{(1)}s_{(3)}\sigma_{ij}\rho_{\alpha}\varphi^{\alpha},\end{aligned}$$

$$\displaystyle\begin{aligned} \widetilde{\mathcal{D}}{}^{(s_{2}).(s_{3}).i}_{jmn}&=\big{(}s_{(2)}\big{)}^{2}\Big{(}F^{i}_{n}\big{(}F^{\alpha}_{m}\sigma_{j}+F^{\alpha}_{j}\sigma_{m}\big{)}\sigma_{\alpha}+F^{i}_{\alpha}F^{\alpha}_{m}\sigma_{j}\sigma_{n}\Big{)}+\big{(}s_{(3)}\big{)}^{2}\sigma_{jm}\sigma_{\alpha n}\varphi^{\alpha}\varphi^{i}\\ &+s_{(2)}s_{(3)}\Big{(}\big{(}F^{\alpha}_{m}\sigma_{j}+F^{\alpha}_{j}\sigma_{m}\big{)}\sigma_{\alpha n}\varphi^{i}-\big{(}F^{i}_{m}\sigma_{\alpha}+F^{i}_{\alpha}\sigma_{m}\big{)}\sigma_{jn}\varphi^{\alpha}\Big{)}\\ &-s_{(2)}\big{(}F^{i}_{j}\sigma_{m}+F^{i}_{m}\sigma_{j}\big{)}_{|n}-s_{(3)}\big{(}\sigma_{jm}\varphi^{i}\big{)}_{|n},\end{aligned}$$

$$\displaystyle(N+1)s_{(1)}\big{(}\overline{\rho}_{j}-\rho_{j}\big{)}$$

$$\displaystyle-L^{\alpha}_{\underline{j\alpha}}+s_{(2)}\big{(}F^{\alpha}_{j}\sigma_{\alpha}+F\sigma_{j}\big{)}+s_{(3)}\sigma_{j\alpha}\varphi^{\alpha},$$

$$\tilde{\overline{T}}{}_{jk}^{(s).i}=\tilde{T}{}_{jk}^{(s).i},$$

$$\tilde{T}{}_{jk}^{(s).i}$$

$$\displaystyle-\frac{s_{(1)}}{N+1}\delta^{i}_{k}\Big{(}L^{\alpha}_{\underline{j\alpha}}-s_{(2)}(F^{\alpha}_{j}\sigma_{\alpha}+F\sigma_{j}\big{)}-s_{(3)}\sigma_{j\alpha}\varphi^{\alpha}\Big{)}$$

$$\displaystyle-s_{(2)}\big{(}F^{i}_{j}\sigma_{k}+F^{i}_{k}\sigma_{j}\big{)}-s_{(3)}\sigma_{jk}\varphi^{i},$$

$$\tilde{T}{}_{jk}^{(s).i}$$

$$\displaystyle+\frac{s_{(1)}}{N+1}\Big{(}s_{(2)}(F^{\alpha}_{k}\sigma_{\alpha}+F\sigma_{k}\big{)}+s_{(3)}\sigma_{k\alpha}\varphi^{\alpha}\Big{)}\delta^{i}_{j}$$

$$\displaystyle+\frac{s_{(1)}}{N+1}\Big{(}s_{(2)}(F^{\alpha}_{j}\sigma_{\alpha}+F\sigma_{j}\big{)}+s_{(3)}\sigma_{j\alpha}\varphi^{\alpha}\Big{)}\delta^{i}_{k},$$

$${\overline{R}}_{jmn}^i$$

$$-\widetilde{\overline{\mathcal{D}}}{}_{jmn}^{(s_2).(s_3).i}+\widetilde{\overline{\mathcal{D}}}{}_{jnm}^{(s_2).(s_3).i}+\tilde{\mathcal{D}}{}_{jmn}^{(s_2).(s_3).i}-\tilde{\mathcal{D}}{}_{jnm}^{(s_2).(s_3).i}.$$

$$\displaystyle(N+1)\big{(}\overline{\zeta}^{(s)}_{[mn]}-\zeta^{(s)}_{[mn]}\big{)}$$

$$+R_{[mn]}-\tilde{\mathcal{D}}{}_{\alpha mn}^{(s_2).(s_3).\alpha }+\tilde{\mathcal{D}}{}_{\alpha nm}^{(s_2).(s_3).\alpha }.$$

$${\overline{R}}_{jmn}^i$$

$$\displaystyle+\frac{1}{N+1}\delta^{i}_{j}\big{(}\widetilde{\overline{\mathcal{D}}}{}^{(s_{2}).(s_{3}).\alpha}_{\alpha mn}-\widetilde{\overline{\mathcal{D}}}{}^{(s_{2}).(s_{3}).\alpha}_{\alpha nm}-\widetilde{{\mathcal{D}}}{}^{(s_{2}).(s_{3}).\alpha}_{\alpha mn}+\widetilde{{\mathcal{D}}}{}^{(s_{2}).(s_{3}).\alpha}_{\alpha nm}\big{)}$$

$$-\widetilde{\overline{\mathcal{D}}}{}_{jmn}^{(s_2).(s_3).i}+\widetilde{\overline{\mathcal{D}}}{}_{jnm}^{(s_2).(s_3).i}+\tilde{\mathcal{D}}{}_{jmn}^{(s_2).(s_3).i}-\tilde{\mathcal{D}}{}_{jnm}^{(s_2).(s_3).i}.$$

$$\displaystyle(N-1)\big{(}\overline{\zeta}^{(s)}_{jm}-\zeta^{(s)}_{jm}\big{)}$$

Full text

Key words: invariant, affine connection coefficients, curvature tensor, Weyl, Thomas

$2010$** Math. Subj. Classification:** 53A55, 53B05, 53C15

1 Introduction

This paper is devoted to generalize the Thomas projective parameter [15] and the Weyl projective tensor [17] as invariants of a mapping of an affine connection space. These invariants are primary generalized in [16] but it is obtained these invariants are not unique ones in this paper. Therefore, we are interested to obtain some invariants of geometric mappings different of the invariants from [16] in this paper.

1.1 Spaces of affine connection

An $N$-dimensional manifold $\mathcal{M}^{N}$ equipped with an affine connection $\nabla$ with torsion is called the non-symmetric affine connection space $\mathbb{GA}_{N}$. The affine connection coefficients of the affine connection $\nabla$ are $L^{i}_{jk}$ and they are non-symmetric by indices $j$ and $k$.

The geometrical object

[TABLE]

satisfies the equation

[TABLE]

so it is the affine connection coefficient of a symmetric affine connection space $\mathbb{A}_{N}$.

The manifold $\mathcal{M}^{N}$ equipped with an affine connection $\widetilde{\nabla}$ whose coefficients are $L^{i}_{\underline{jk}}$ is the associated space $\mathbb{A}_{N}$ (of the space $\mathbb{GA}_{N}$).

There is one kind of covariant derivation with regard to a symmetric affine connection:

[TABLE]

for a tensor $a^{i}_{j}$ of the type $(1,1)$ and the partial derivative $\partial/\partial x^{i}$ denoted by comma.

From the corresponding Ricci-type identity, it is obtained one curvature tensor of the associated space $\mathbb{A}_{N}$:

[TABLE]

Special symmetric affine connection spaces are Riemannian spaces $\mathbb{R}_{N}$ whose affine connection coefficients are the corresponding Christoffel symbols $\Gamma^{i}_{\underline{jk}}$ of the second kind.

Many authors have developed the theory of symmetric affine connection spaces and mappings between them. Some of them are J. Mikeš [9, 2, 12, 8, 11, 10, 1, 3], N. S. Sinyukov [13], V. E. Berezovski [9, 11, 2, 1, 3], L. P. Eisenhart [4] and many others. M. S. Stanković (see [14]) obtained an invariant of an almost geodesic mapping of a non-symmetric affine connection space from the corresponding transformation of the curvature tensor (1.3).

1.2 Motivation

H. Weyl [17] and T. Y. Thomas [15] obtained invariants of geodesic mappings of a symmetric affine connection space. J. Mikeš with his research group [9, 2, 12, 8, 11, 10, 1, 3], N. S. Sinyukov [13], and many other authors have continued the process of generalization of these invariants.

The search for general formula for invariants of geometric mappings is started in [16]. The basic invariants of geometric mappings are obtained in this paper. It is also studied a special case in that paper. In this special case, it is obtained basic invariants of a mapping but the author founded some other invariants different of the basic ones. We are interested to obtain some of these invariants which are not basic in this paper.

The invariants which we are interested to develop in this paper are

• •

The generalized Thomas projective parameter [15]:

[TABLE]

• •

The Weyl projective tensor [17]:

[TABLE]

In a Riemannian space $\mathbb{R}_{N}$, the Weyl projective tensor (1.5) reduces to

[TABLE]

2 Reminder on basic invariants

Let $f:\mathbb{GA}_{N}\to\mathbb{G\overline{A}}_{N}$ be a mapping between non-symmetric affine connection spaces $\mathbb{GA}_{N}$ and $\mathbb{G\overline{A}}_{N}$. The deformation tensor $P^{i}_{jk}$ of this mapping satisfies the corresponding equation [16]

[TABLE]

for geometrical objects $\omega^{i}_{jk},\overline{\omega}{}^{i}_{jk},\tau^{i}_{jk},\overline{\tau}{}^{i}_{jk}$ of the type $(1,2)$ such that $\omega^{i}_{jk}=\omega^{i}_{kj},$ $\overline{\omega}{}^{i}_{jk}=\overline{\omega}{}^{i}_{kj},$ $\tau^{i}_{jk}=-\tau^{i}_{kj},$ $\overline{\tau}{}^{i}_{jk}=-\overline{\tau}{}^{i}_{kj}$.

After symmetrizing the equation (2.1) by indices $j$ and $k$, one gets

[TABLE]

With regard to the last result, it is obtained three kinds of invariants of the mapping $f$ in [16]. In this paper, we are interested to generalize just invariants of the second kind. The basic invariants which we will generalize are:

[TABLE]

for $p^{1}_{1},\ldots,p^{2}_{3}=1,2,\omega_{(1).jk}^{i}=L^{i}_{\underline{jk}},\omega^{i}_{(2).jk}=\omega^{i}_{jk}$. The invariant (2.3) is the basic invariant of the mapping $f$ of the Thomas type but the invariant (2.4) is the basic invariant of the mapping $f$ of the Weyl type.

3 Derived invariants

In general, the geometrical object $\omega^{i}_{jk}$ from the equation (2.1) has the form

[TABLE]

for $s_{1},s_{2},s_{3}\in\mathbb{R}$, $1$-forms $\rho_{j},\sigma_{j}$, an affinor $F^{i}_{j}$, a covariant tensor $\sigma_{jk}$ symmetric by indices $j$ and $k$ and a contra-variant vector $\varphi^{i}$.

Remark 3.1**.**

The geometrical objects $\delta^{i}_{j}\rho_{k}+\delta^{i}_{k}\rho_{j}$, $F^{i}_{j}\sigma_{k}+F^{i}_{k}\sigma_{j}$, $\sigma_{jk}\varphi^{i}$ are linearly independent. Otherwise, there would not be necessary three constants $s_{(1)},s_{(2)},s_{(3)}$.

With regard to the equation (3.1), we get

[TABLE]

It holds the following theorem:

Theorem 3.1**.**

Let $f:\mathbb{GA}_{N}\to\mathbb{G\overline{A}}_{N}$ be a geometric mapping. The geometrical objects

[TABLE]

for

[TABLE]

$s=(s_{1},s_{2},s_{3})$, are the basic invariants of the mapping $f$. ∎

3.1 Invariants in associated space

We will analyze the invariants (3.3, 3.4) of a mapping $f:\mathbb{GA}_{N}\to\mathbb{G\overline{A}}_{N}$ bellow. From this analyzing, we will obtain some other invariants of the mapping $f$.

After contracting the equality $\widetilde{\overline{\mathcal{T}}}{}^{i}_{jk}-\widetilde{\mathcal{T}}{}^{i}_{jk}=0$ by indices $i$ and $k$, we get

[TABLE]

for $F=F^{\alpha}_{\alpha},\overline{F}=\overline{F}^{\alpha}_{\alpha}$.

After substituting this equation into the equality $\widetilde{\overline{\mathcal{T}}}{}^{(s).i}_{jk}-\widetilde{\mathcal{T}}{}^{(s).i}_{jk}=0$, one obtains

[TABLE]

for

[TABLE]

and the corresponding $\widetilde{\overline{T}}{}^{(s).i}_{jk}$.

Lemma 3.1**.**

Let $f:\mathbb{GA}_{N}\to\mathbb{G\overline{A}}_{N}$ be a geometric mapping of a non-symmetric affine connection space $\mathbb{GA}_{N}$. The geometrical object (3.8) is an invariant of the mapping $f$. ∎

Corollary 3.1**.**

The invariant (3.8) and the Thomas projective parameter $T^{i}_{jk}$ given in the equation (1.4) satisfy the equation

[TABLE]

for the corresponding $s=(s_{1},s_{2},s_{3})$. ∎

The geometrical object (3.8) is the derived associated invariant of the Thomas type.

Furthermore, from the invariance of the geometrical object (3.4) we obtain

[TABLE]

After contracting this equation by indices $i$ and $j$, we get

[TABLE]

From the equations (3.10, 3.11), one obtains

[TABLE]

From the contraction of this result by indices $i$ and $n$, we get

[TABLE]

With regard to the equations (3.12, 3.13), we get

[TABLE]

for

[TABLE]

and the corresponding ${{\overline{W}}}{}^{(s).[1].i}_{jmn}$.

Let us test are some summands in the invariant ($i$) invariants of the mapping $f$. After contract the equality ${{\overline{W}}}{}^{(s).[1].i}_{jmn}-{{W}}{}^{(s).[1].i}_{jmn}=0$ by indices $i$ and $n$, we get

[TABLE]

Hence, the invariant ($i$) reduces to

[TABLE]

and the corresponding ${\overline{W}}{}^{(s).[2].i}_{jmn}$.

Let us check-out are there invariants of the mapping $f$ into the invariant ($ii$). After contracting the equality ${\overline{W}}{}^{(s).[2].i}_{jmn}-{{W}}{}^{(s).[2].i}_{jmn}=0$ by the indices $i$ and $j$, one obtains

[TABLE]

Hereof, the invariant ($ii$) reduces to

[TABLE]

It holds the following theorem:

Theorem 3.2**.**

Let $f:\mathbb{GA}_{N}\to\mathbb{G\overline{A}}_{N}$ be a geometric mapping. The geometrical object(3.14)* is an invariant of this mapping. ∎*

Corollary 3.2**.**

The invariant (3.14) and the Weyl projective tensor (1.5) satisfy the equation

[TABLE]

for the corresponding $s=(s_{1},s_{2},s_{3})$. ∎

The geometrical object (3.14) is the derived associated invariant of the Weyl type.

3.2 $F$-planar mappings

In this part of paper, we will apply the above obtained results. The theoretical part of this application will be search for invariants of conformal mappings. The practical example will be about a transformation of a special Riemannian space.

$F$-planar mappings. A mapping $f:\mathbb{A}_{N}\to\mathbb{\overline{A}}_{N}$ is called the $F$-planar mapping if it is determined with the equation

[TABLE]

for $1$-forms $\psi_{j},\sigma_{j}$ and an affinor $F^{i}_{j}$.

The basic equation of the inverse mapping $f^{-1}:\mathbb{A}_{N}\to\mathbb{\overline{A}}_{N}$ is

[TABLE]

Hence, we get the mapping $f^{-1}$ is an $F$-planar mapping for

[TABLE]

Moreover, it holds

[TABLE]

Therefore, the $F$-planar mapping $f$ is the case of $s_{1}=1,s_{2}=1/2,s_{3}=0$, in the equation (3.1).

After contracting the equation (3.19) by indices $i$ and $k$, one gets

[TABLE]

With regard to the equations (3.1, 3.19, 3.20), we obtain

[TABLE]

for $F=F^{\alpha}_{\alpha}$.

From the equation (3.3), it holds that the associated invariant of the Thomas type of the mapping $f$ is

[TABLE]

for the generalized Thomas projective parameter $T^{i}_{jk}$ from the equation (1.4).

Based on the equation (3.18), we obtain

[TABLE]

i.e. the geometrical objects (3.5, 3.6) reduce to

[TABLE]

Therefore, the invariant (3.4) of the mapping $f$ is

[TABLE]

for $\zeta_{ij}$ given in the equation (3.24).

The invariant (3.14) of the $F$-planar mapping $f$ is

[TABLE]

for the Weyl projective tensor $W^{i}_{jmn}$.

Example 1**.**

In this example, we are aimed to obtain the invariants (3.22, 3.25, 3.26) of an $F$-planar mapping $f:\mathbb{R}_{3}\to\mathbb{\overline{R}}_{3}$ for the Riemannian space $\mathbb{R}_{3}$ determined with the metric tensor

[TABLE]

The corresponding affinor $F^{i}_{j}$ and covariant vector $\sigma_{j}$ are

[TABLE]

Let be $\mathcal{F}^{i}_{jk}=F^{i}_{k}\sigma_{j}+F^{i}_{j}\sigma_{k}$. It is satisfied

[TABLE]

It also holds

[TABLE]

The Christoffel symbols of the second kind of this space are

[TABLE]

and $\Gamma^{i}_{\underline{jk}}=0$ in all other cases.

The generalized Thomas projective parameter of the space $\mathbb{R}_{3}$ is

[TABLE]

Hence, the derived invariant of Thomas type of the mapping $f$ is

[TABLE]

for $\mathcal{F}^{i}_{jk},\mathcal{F}^{\alpha}_{j\alpha}$ given by the equations (3.35, 3.36).

We have the following cases for the curvature tensor $R^{i}_{jmn}$:

[TABLE]

It is also satisfied

[TABLE]

for $\mathcal{F}^{i}_{jk},\mathcal{F}^{\alpha}_{j\alpha}$ given in the equations (3.35, 3.36).

Hence, the corresponding invariants of Weyl type of the mapping $f$ are

[TABLE]