On ($\overline{\mathit{m}}$, m)-Conformal ($\overline{\mathit{F}}$, F)-Mappings

Abstract

This paper investigates novel transformations of affine connections and the derivation of their corresponding invariants within the framework of differential geometry. Drawing inspiration from previously established $(\overline{m},m)$-conformal mappings, we introduce a generalized class of these transformations. We successfully derive three fundamental invariants for this class. To demonstrate the geometric significance of these structures, we examine their implications within the calculus of variations. Finally, we provide potential physical interpretations of the obtained results, suggesting their relevance in theoretical physics.

1. Introduction

This research is focused on generalizing of Einstein–Hilbert actions used in cosmology. We aim to develop the standard approach of deriving Einstein’s equation by defining a novel class of geometric mappings and obtaining invariants for them.

1.1. Abstraction by Differential Geometry of Mappings

Differential geometry is a branch of mathematics fundamentally connected with various sub-disciplines in physics. For the purposes of our research, it is necessary to establish several basic definitions of geometric terms, which will be reviewed based on the frameworks provided in [1,2].

An N-dimensional manifold $\mathcal{M}=\mathcal{M}\left(x\right)$ equipped with a symmetric covariant metric tensor $g_{\alpha \beta }$ is an N-dimensional Riemannian space ${\mathbb{R}}_N$. We assume that the matrix $\left[g_{\alpha \beta }\right]$ is non-singular, i.e., $\det \left[g_{\alpha \beta }\right]\ne 0$. Consequently, the contravariant metric tensor is defined as $\left[g^{\alpha \beta }\right]={\left[g_{\alpha \beta }\right]}^{-1}$.

In a curved space, the geometry is characterized by the affine connection coefficients. For a Riemannian space, these coefficients are the Christoffel symbols of the second kind

$${\Gamma }_{\mu \nu }^{\pi }={\displaystyle \frac{1}{2}}{g}^{\pi \alpha }\left(g_{\mu \alpha ,\nu }-g_{\mu \nu ,\alpha }+g_{\nu \alpha ,\mu }\right),$$

where the comma denotes partial differentiation ($g_{\pi \mu ,\nu }=\partial g_{\pi \mu }/\partial x^{\nu }$). The Einstein summation convention is applied to the repeated (mute) index $\alpha$, ranging from 0 to $N-1$ for Riemannian spacetime, or from 1 to N for Riemannian space.

Under a coordinate transformation $x\to x^{\prime }$, the Christoffel symbols transform according to

$${\Gamma }_{{\mu }^{\prime }{\nu }^{\prime }}^{{\pi }^{\prime }}=x^{{\pi }^{\prime }}\pi x_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{\Gamma }_{\mu \nu }^{\pi }+x^{{\pi }^{\prime }}\pi x^{\pi }{\mu }^{\prime }{\nu }^{\prime },$$

(1)

where $x_{\pi }^{{\pi }^{\prime }}={\displaystyle \frac{\partial x^{\prime \pi }}{\partial x^{\pi }}}$, $x_{{\mu }^{\prime }}^{\mu }={\displaystyle \frac{\partial x^{\mu }}{\partial x^{\prime \mu }}}$, and $x_{{\mu }^{\prime }{\nu }^{\prime }}^{\pi }={\displaystyle \frac{{\partial }^2{x}^{\pi }}{\partial x^{\prime \mu }\partial x^{\prime \nu }}}$.

A multi-indexed object $t_{\mu }^{\pi }$ is defined as a $(1,1)$-type tensor if its transformation law follows

$$t_{{\mu }^{\prime }}^{{\pi }^{\prime }}=x^{{\pi }^{\prime }}\pi x_{{\mu }^{\prime }}^{\mu }{t}_{\mu }^{\pi }.$$

(2)

The covariant derivative with respect to the affine connection ${\Gamma }_{\mu \nu }^{\pi }$ of the tensor $t_{\beta }^{\alpha }$, which preserves the tensor characteristic of it as

$$t_{\beta |\gamma }^{\alpha }={\tau }_{\beta ,\gamma }^{\alpha }+{\Gamma }_{\delta \gamma }^{\alpha }{t}_{\beta }^{\delta }-{\Gamma }_{\beta \gamma }^{\delta }{t}_{\delta }^{\alpha }.$$

By comparing (1) and (2), it is evident that Christoffel symbols are not tensors. However, the difference between two sets of coefficients for different affine connections, known as the deformation tensor

$$P_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi },$$

is a tensor of type $(1,2)$. If ${\mathbb{R}}_N$ and ${\overline{\mathbb{R}}}_N$ are Riemannian spaces, the transformation ${\overline{\Gamma }}_{\mu \nu }^{\pi }\to {\Gamma }_{\mu \nu }^{\pi }$ can be categorized as:

• Geodesic, if ${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }$ for a 1-form ${\psi }_{\mu }$.
• Conformal, if $g_{\mu \nu }=e^{2\psi }{\overline{g}}_{\mu \nu }$, which is equivalent to ${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }-{\psi }_{\alpha }{g}^{\pi \alpha }{g}_{\mu \nu }$.
• F-planar, if ${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }+{\sigma }_{\nu }{F}_{\mu }^{\pi }+{\sigma }_{\mu }{F}_{\nu }^{\pi }$ for 1-forms ${\psi }_{\mu },{\sigma }_{\mu }$ and a tensor of the type $(1,1)$ (affinor) $F_{\mu }^{\pi }$.

While numerous other mappings exist, these examples serve as a baseline for current research. The curvature tensor associated with the affine connection ${\Gamma }_{\mu \nu }^{\pi }$ is defined as

$$R_{\mu \nu \sigma }^{\pi }={\Gamma }_{\mu \nu ,\sigma }^{\pi }-{\Gamma }_{\mu \sigma ,\nu }^{\pi }+{\Gamma }_{\mu \nu }^{\alpha }{\Gamma }_{\alpha \sigma }^{\pi }-{\Gamma }_{\mu \sigma }^{\alpha }{\Gamma }_{\alpha \nu }^{\pi }.$$

(3)

The corresponding Ricci tensor, obtained by the contraction $R_{\mu \sigma }=R_{\mu \alpha \sigma }^{\alpha }$, is

$$R_{\mu \sigma }={\Gamma }_{\mu \alpha ,\sigma }^{\alpha }-{\Gamma }_{\mu \sigma ,\alpha }^{\alpha }+{\Gamma }_{\mu \alpha }^{\beta }{\Gamma }_{\sigma \beta }^{\alpha }-{\Gamma }_{\mu \sigma }^{\alpha }{\Gamma }_{\alpha },$$

(4)

where ${\Gamma }_{\alpha }={\Gamma }_{\alpha \beta }^{\beta }$. The scalar curvature $R=g^{\alpha \beta }{R}_{\alpha \beta }$ follows as

$$R=g^{\alpha \beta }{\Gamma }_{\alpha \gamma ,\beta }^{\gamma }-g^{\alpha \beta }{\Gamma }_{\alpha \beta ,\gamma }^{\gamma }+g^{\alpha \beta }{\Gamma }_{\alpha \delta }^{\gamma }{\Gamma }_{\beta \gamma }^{\delta }-g^{\alpha \beta }{\Gamma }_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }.$$

(5)

A central problem in differential geometry is the derivation of geometric objects expressed in terms of the curvature tensor (3), Ricci tensor (4), scalar curvature (5), and auxiliary fields ($F_{\mu }^{\pi },{\sigma }_k$) that remain invariant under the transformation ${\overline{\Gamma }}_{\mu \nu }^{\pi }\to {\Gamma }_{\mu \nu }^{\pi }$.

H. Weyl proposed a method for obtaining the invariant of a conformal mapping, known as the trace-free Weyl conformal tensor [3]. Similarly, T. Thomas derived the Thomas projective parameter as an invariant for geodesic mappings [4]. Following these methodologies, various authors have derived invariants for different mappings [5,6,7,8,9,10,11], typically of the form $W_{\mu \nu \sigma }^{\pi }=R_{\mu \nu \sigma }^{\pi }+X_{\mu \nu \sigma }^{\pi }$, where the Ricci tensor $R_{\mu \sigma }$ is eliminated in any contraction (e.g., $W_{\mu \alpha \sigma }^{\alpha }=0$).

The Weyl methodology was further expanded in [12,13], leading to the identification of two distinct invariants: ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and $W_{\mu \nu \sigma }^{\pi }$. Unlike traditional invariants, the trace ${\mathcal{W}}_{\mu \sigma }={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }$ does not identically vanish. Conversely, the invariant $W_{\mu \nu \sigma }^{\pi }$ follows the classical trace-free property, effectively vanishing the Ricci tensor from its contraction $W_{\mu \alpha \sigma }^{\alpha }$.

This leads to the central question of this research: can the invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and its non-vanishing trace ${\mathcal{W}}_{\mu \sigma }$ be effectively integrated into physical theories?

1.2. Physical Concretization of Differential Geometry

The standard approach to deriving field equations begins with the Einstein–Hilbert action [14], $\int d^{4}x\sqrt{-g}R$. By requiring its variation to vanish, the well-known energy-momentum tensor relation, $2\kappa T_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }$, is obtained. Subsequent generalizations of the scalar curvature R have led to the development of various alternative cosmological theories.

Starting from initial Einstein–Maxwell action $\int d^{4}x\sqrt{-g}\left(g^{\alpha \beta }{R}_{\alpha \beta }+F^{\alpha \beta }{F}_{\alpha \beta }\right)$, different authors created the developed models. These authors include Mena and co-workers [15], who investigated and stability of special space-times, Razina and co-authors [16], who added a new element to the starting Einstein–Maxwell action for suitability of research, and Alpin and Balakin [17], who formulated a modified Einstein–Maxwell model that may be applied to different interactions with electromagnetically active material media.

Remark 1.

The affinor $F_{\beta }^{\alpha }$ in the space ${\mathbb{R}}_N$ may be presented as $F_{\beta }^{\alpha }=g^{\alpha \gamma }{F}_{\gamma \beta }$. In the same manner, $F^{\alpha \beta }=g^{\alpha \gamma }{g}^{\beta \delta }{F}_{\gamma \delta }$, and $F_{\alpha \beta }=g_{\alpha \gamma }{F}_{\beta }^{\gamma }$. If we assume $F_{\alpha \beta }=-F_{\beta \alpha }$, the next relations will be satisfied

$$F_{\beta }^{\alpha }{F}_{\alpha }^{\beta }=g^{\alpha \gamma }{F}_{\gamma \beta }{g}^{\beta \delta }{F}_{\delta \alpha }=F_{\gamma \beta }{g}^{\alpha \gamma }{g}^{\beta \delta }{F}_{\delta \alpha }=F_{\beta \gamma }{F}^{\gamma \beta }=-F_{\beta \gamma }{F}^{\beta \gamma }.$$

In the case of $N=4$, we identify the composition $F^{\alpha \beta }{F}_{\alpha \beta }$ in the Einstein–Maxwell model with negative composition of affinors $F_{\beta }^{\alpha }{F}_{\alpha }^{\beta }$ such that $F_{\alpha \beta }=-F_{\beta \alpha }$.

In the realm of quantum gravity, D. G. C. McKeon [18] investigated the quantization of the first-order two-dimensional Einstein–Hilbert action. By re-parameterizing the scalar curvature of a four-dimensional Riemannian space, the research facilitated a framework for action quantization. Similarly, S. Chackraborti [19] explored extensions of Einstein’s classical model by expanding the dimensionality to $D>4$ and incorporating the Gibbons–Hawking–York boundary term [20,21,22] to ensure a well-defined variational problem.

The transformation of the metric tensor and its impact on the resulting Einstein–Hilbert action was analyzed by A. Huber [23], focusing on how transformed metrics alter the scalar curvature. In the field of growth mechanics, Yavari [24] employed conformal mappings of two- and three-dimensional Riemannian spaces to transform the metric tensor. While that study discussed the conditions under which the variation of the Ricci tensor may be neglected, it did not extend the analysis to the corresponding Weyl tensors as invariants for such transformations.

Furthermore, M. Minozzi et al. [25] addressed time-evolution problems involving distortion fields that generate non-Euclidean metrics within spherical domains. These diverse approaches demonstrate that different geometric structures are tailored to specific physical problems. While cosmological models rely heavily on Ricci tensors and scalar curvatures, applications in mechanics—such as those by Yavari and Minozzi—utilize these terms differently or focus on localized distortions.

This provides the motivation for our current study: to present a unified geometric model whose variation is equally applicable to both cosmology and classical mechanics, depending on the chosen dimensionality. Our model is fundamentally based on a specialized transformation of Christoffel symbols under perturbation, bridging the gap between abstract mappings and concrete physical dynamics.

1.3. Motivation: Aims and Scope of This Research

In standard General Relativity, test particles follow geodesic lines in the absence of non-gravitational forces. However, in most physical scenarios, additional interactions influence the motion, causing trajectories to deviate from purely geodesic paths. This research introduces a framework where these additional effects are integrated directly into the geometry of the manifold. We define a new class of trajectories, transitioning from standard geodesics to specialized curves that encapsulate these interactions.

The purposes of this study are as follows:

1.

To generalize the concept of $(\overline{m},m)$-lines [26] into a broader class of $(\overline{m},m)$-conformal $(\overline{F},F)$-lines. This novel class of curves serves as the foundation for defining a new category of geometric mappings.

2.

To apply a rigorous methodology for deriving invariants associated with these mappings. We aim to identify a non-trivial invariant related to the Ricci tensor, as well as a $(1,3)$-type invariant analogous to the Weyl tensor.

3.

To perform a variational analysis of the action $\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{\mathcal{W}}_{\alpha \beta }$, where ${\mathcal{W}}_{\alpha \beta }$ is a monic polynomial invariant derived from the mapping. This process will yield a generalized form of Einstein’s field equations.

4.

To construct analogies of gravitoelectric and gravitomagnetic tensors based on the trace-free invariant obtained in the previous steps.

The overarching goal of this article is to demonstrate how the differential geometry of mappings and perturbation theory in physics can be unified through a single geometric object. We show that traditional tensor characteristics emerge as special cases of our framework, thereby supporting and extending Einstein’s fundamental results from 1916 [27]. Furthermore, this generalization offers significant applications in both classical and continuum mechanics, and in cosmology with respect to different dimensional Riemannian space-times.

This article is composed as below:

1.

In Section 2, the methodology of obtaining invariants for geometric mappings will reviewed.

2.

In Section 3, the concept of geodesic lines will be generalized and a novel class of mappings will be defined.

3.

In Section 4, the invariants for mappings from novel class will be obtained.

4.

In Section 5, the variation of different geometrical objects constructed by invariants obtained in Section 4 will be perturbed, which will result in the corresponding Einstein’s equations and other interpretations analogous to the physical ones.

5.

In Section 6, we will explain the analogies between our theoretical results and different applications in physics.

2. Review on Invariants for Geometric Mappings

In the research from 2020 and 2022 [12,13], the question was whether Weyl’s methodology for obtaining invariants of geometric mappings can be developed in any way. Let us review this methodology.

If $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ is a geodesic mapping between Riemannian spaces, their affine connection coefficients (which are Christoffel symbols) satisfy the equality

$${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi },$$

(6)

where ${\psi }_{\mu }$ is a 1-form.

After contracting this equation by $\pi$ and $\nu$, one obtains

$${\psi }_{\mu }={\displaystyle \frac{1}{N+1}}\left({\Gamma }_{\mu }-{\overline{\Gamma }}_{\mu }\right),$$

for ${\Gamma }_{\mu }={\Gamma }_{\mu \alpha }^{\alpha }$ and ${\overline{\Gamma }}_{\mu }={\overline{\Gamma }}_{\mu \alpha }^{\alpha }$, which substituted in (6) gives

$${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu }\right)-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\overline{\Gamma }}_{\nu }+{\delta }_{\nu }^{\pi }{\overline{\Gamma }}_{\mu }\right).$$

(7)

The relation (7) is equivalent to $G{T}_{\mu \nu }^{\pi }=G{\overline{T}}_{\mu \nu }^{\pi }$, for

$$G{T}_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu }\right),$$

(8)

and the corresponding $G{\overline{T}}_{\mu \nu }^{\pi }$.

In this way, it was proved that the geometrical object $G{T}_{\mu \nu }^{\pi }$, known as the Thomas projective parameter, is an invariant for the geodesic mapping f.

It also holds that a mapping $\varphi :{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ is geodesic if and only if the Thomas projective parameter given by (8) stays unchanged under application $\varphi$ on it.

The transformation rule of curvature tensor ${\overline{R}}_{\mu \nu \sigma }^{\pi }$ to $R_{\mu \nu \sigma }^{\pi }$ is

$$R_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }+{\delta }_{\mu }^{\pi }{\psi }_{[\nu ,\sigma ]}+{\delta }_{\nu }^{\pi }({\psi }_{\mu ,\sigma }-{\psi }_{\mu }{\psi }_{\sigma })-{\delta }_{\sigma }^{\pi }({\psi }_{\mu ,\nu }-{\psi }_{\mu }{\psi }_{\nu }),$$

(9)

where square brackets denote antisymmetrization without division.

After contracting the relation (9) by $\pi$ and $\mu$, such as by $\pi$ and $\nu$, one obtains the following relations

$$GS:\left\{\begin{array}{c}0=(N+1){\psi }_{[\nu ,\sigma ]}\hfill \\ R_{\mu \sigma }={\overline{R}}_{\mu \sigma }+{\psi }_{[\mu ,\sigma ]}+(N-1)({\psi }_{\mu ,\sigma }-{\psi }_{\mu }{\psi }_{\sigma }).\hfill \end{array}\right.$$

(10)

The solutions of system GS given by (10) are ${\psi }_{[\nu ,\sigma ]}=0$ and ${\psi }_{\mu ,\sigma }-{\psi }_{\mu }{\psi }_{\sigma }={\displaystyle \frac{1}{N-1}}\left(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }\right)$. After substituting this solution into the transformation rule (9), we obtain

$$R_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }+{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{R}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\right)-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{\overline{R}}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{\overline{R}}_{\mu \nu }\right).$$

(11)

The relation (11) is equivalent to $G{W}_{\mu \nu \sigma }^{\pi }=G{\overline{W}}_{\mu \nu \sigma }^{\pi }$, for

$$G{W}_{\mu \nu \sigma }^{\pi }=R_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{R}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\right),$$

(12)

and the corresponding $G{\overline{W}}_{\mu \nu \sigma }^{\pi }$.

This proves that the geometrical object $G{W}_{\mu \nu \sigma }^{\pi }$, well known as the Weyl projective tensor, is an invariant for the geodesic mapping f.

This method of obtaining of $G{W}_{\mu \nu \sigma }^{\pi }$ as an invariant for the geodesic mapping f means that the invariant $G{W}_{\mu \nu \sigma }^{\pi }$ is a generalization of the invariant $G{T}_{\mu \nu }^{\pi }$. From the other hand, this generalization is not direct.

The direct generalization of $G{T}_{\mu \nu }^{\pi }$ to an invariant of geodesic mapping f is obtained in [13] from the equality

$$\begin{array}{cc}\hfill G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }& =G{T}_{\mu \nu ,\sigma }^{\pi }-G{T}_{\mu \sigma ,\nu }^{\pi }+G{T}_{\mu \nu }^{\alpha }G{T}_{\alpha \sigma }^{\pi }-G{T}_{\mu \sigma }^{\alpha }G{T}_{\alpha \nu }^{\pi }\hfill \\ & =G{\overline{T}}_{\mu \nu ,\sigma }^{\pi }-G{\overline{T}}_{\mu \sigma ,\nu }^{\pi }+G{\overline{T}}_{\mu \nu }^{\alpha }G{\overline{T}}_{\alpha \sigma }^{\pi }-G{\overline{T}}_{\mu \sigma }^{\alpha }G{\overline{T}}_{\alpha \nu }^{\pi }=:G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }.\hfill \end{array}$$

(13)

After inserting the definition (8) of Thomas projective parameter $G{T}_{\mu \nu }^{\pi }$, and its image $G{\overline{T}}_{\mu \nu }^{\pi }$ under geodesic mapping f, into the relation (13), one obtains

$$\begin{array}{cc}\hfill G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }& =R_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\nu }^{\pi }\left((N+1){\Gamma }_{\mu |\sigma }+{\Gamma }_{\mu }{\Gamma }_{\sigma }\right)\hfill \\ & +{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\sigma }^{\pi }\left((N+1){\Gamma }_{\mu |\nu }+{\Gamma }_{\mu }{\Gamma }_{\nu }\right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }& ={\overline{R}}_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\nu }^{\pi }\left((N+1){\overline{\Gamma }}_{\mu \parallel \sigma }+{\overline{\Gamma }}_{\mu }{\overline{\Gamma }}_{\sigma }\right)\hfill \\ & +{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\sigma }^{\pi }\left((N+1){\overline{\Gamma }}_{\mu \parallel \nu }+{\overline{\Gamma }}_{\mu }{\overline{\Gamma }}_{\nu }\right),\hfill \end{array}$$

(15)

for the covariant derivative with respect to ${\overline{\Gamma }}_{\mu \nu }^{\pi }$ denoted by "‖".

The invariants $G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and $G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ are expressed in the forms $G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }=R_{\mu \nu \sigma }^{\pi }-{\delta }_{\nu }^{\pi }{X}_{\mu \sigma }+{\delta }_{\sigma }^{\pi }{X}_{\mu \nu }$ and $G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }-{\delta }_{\nu }^{\pi }{\overline{X}}_{\mu \sigma }+{\delta }_{\sigma }^{\pi }{\overline{X}}_{\mu \nu }$, where $X_{\mu \nu }={\displaystyle \frac{1}{{(N+1)}^2}}\left((N+1){\Gamma }_{\mu |\nu }+{\Gamma }_{\mu }{\Gamma }_{\nu }\right)$ and ${\overline{X}}_{\mu \nu }={\displaystyle \frac{1}{{(N+1)}^2}}((N+1){\overline{\Gamma }}_{\mu \parallel \nu }+{\overline{\Gamma }}_{\mu }{\overline{\Gamma }}_{\nu }$. The difference $0=G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }-G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ has the form $0=\left(R_{\mu \nu \sigma }^{\pi }-{\overline{R}}_{\mu \nu \sigma }^{\pi }\right)-{\delta }_{\nu }^{\pi }\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right)+{\delta }_{\sigma }^{\pi }\left(X_{\mu \nu }-{\overline{X}}_{\mu \nu }\right)$. After contracting this equality by $\pi$ and $\nu$, we obtain $X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }={\displaystyle \frac{1}{N-1}}(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }$, which transforms the equality $0=G{\mathcal{W}}_{\mu \nu \sigma }^{\pi }-G{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ to

$$0=\left(R_{\mu \nu \sigma }^{\pi }-{\overline{R}}_{\mu \nu \sigma }^{\pi }\right)-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{R}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\right)+{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{\overline{R}}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{\overline{R}}_{\mu \nu }\right),$$

which is equivalent to the equality of Weyl projective tensors of the spaces ${\mathbb{R}}_N$ and ${\overline{R}}_N$, $G{W}_{\mu \nu \sigma }^{\pi }=G{\overline{W}}_{\mu \nu \sigma }^{\pi }$.

Motivated by previous methodology, for a mapping $\phi :{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$, the deformation tensor $P_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }$ was expressed in the form

$$P_{\mu \nu }^{\pi }={\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }+{\omega }_{\mu \nu }^{\pi }-{\overline{\omega }}_{\mu \nu }^{\pi },$$

(16)

where ${\omega }_{\mu \nu }^{\pi }$ and ${\overline{\omega }}_{\mu \nu }^{\pi }$ are geometrical objects of the type $(1,2)$ symmetric by $\mu$ and $\nu$.

The contraction of relation (16) by $\pi$ and $\nu$ gives

$${\psi }_{\mu }={\displaystyle \frac{1}{N+1}}\left({\Gamma }_{\mu }-{\omega }_{\mu \alpha }^{\alpha }\right)-{\displaystyle \frac{1}{N+1}}\left({\overline{\Gamma }}_{\mu }-{\overline{\omega }}_{\mu \alpha }^{\alpha }\right).$$

(17)

After substituting the expression (17) of ${\psi }_{\mu }$ into the relation (16), we obtain

$$\begin{array}{cc}\hfill P_{\mu \nu }^{\pi }& ={\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\Gamma }_{\nu }-{\omega }_{\nu \alpha }^{\alpha }\right)+{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left({\Gamma }_{\mu }-{\omega }_{\mu \alpha }^{\alpha }\right)+{\omega }_{\mu \nu }^{\pi }\hfill \\ & -{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\overline{\Gamma }}_{\nu }-{\overline{\omega }}_{\nu \alpha }^{\alpha }\right)-{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left({\overline{\Gamma }}_{\mu }-{\overline{\omega }}_{\mu \alpha }^{\alpha }\right)-{\overline{\omega }}_{\mu \nu }^{\pi }.\hfill \end{array}$$

(18)

Based on the definition of deformation tensor $P_{\mu \nu }^{\pi }$, the relation (18) is equivalent to ${\mathcal{T}}_{\mu \nu }^{\pi }={\overline{\mathcal{T}}}_{\mu \nu }^{\pi }$, for

$${\mathcal{T}}_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\omega }_{\mu \nu }^{\pi }-{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\Gamma }_{\nu }-{\omega }_{\nu \alpha }^{\alpha }\right)-{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left({\Gamma }_{\mu }-{\omega }_{\mu \alpha }^{\alpha }\right),$$

(19)

and the corresponding ${\overline{\mathcal{T}}}_{\mu \nu }^{\pi }$.

From the equality

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mu \nu \sigma }^{\pi }& :={\mathcal{T}}_{\mu \nu ,\sigma }^{\pi }-{\mathcal{T}}_{\mu \sigma ,\nu }^{\pi }+{\mathcal{T}}_{\mu \nu }^{\alpha }{\mathcal{T}}_{\alpha \sigma }^{\pi }-{\mathcal{T}}_{\mu \sigma }^{\alpha }{\mathcal{T}}_{\alpha \nu }^{\pi }\hfill \\ & ={\overline{\mathcal{T}}}_{\mu \nu ,\sigma }^{\pi }-{\overline{\mathcal{T}}}_{\mu \sigma ,\nu }^{\pi }+{\overline{\mathcal{T}}}_{\mu \nu }^{\alpha }{\overline{\mathcal{T}}}_{\alpha \sigma }^{\pi }-{\overline{\mathcal{T}}}_{\mu \sigma }^{\alpha }{\overline{\mathcal{T}}}_{\alpha \nu }^{\pi }=:{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi },\hfill \end{array}$$

one gets the equality ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }={\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ for

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mu \nu \sigma }^{\pi }& =R_{\mu \nu \sigma }^{\pi }-{\omega }_{\mu \nu |\sigma }^{\pi }+{\omega }_{\mu \sigma |\nu }^{\pi }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha \sigma }^{\pi }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }{\omega }_{\left[\nu \alpha \right|\sigma ]}^{\alpha }\hfill \\ & -{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\nu }^{\pi }\left((N+1)\left({\Gamma }_{\mu |\sigma }-{\omega }_{\mu \alpha |\sigma }^{\alpha }+{\omega }_{\mu \sigma }^{\alpha }({\Gamma }_{\alpha }-{\omega }_{\alpha \beta }^{\beta })\right)+\left({\Gamma }_{\mu }-{\omega }_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\sigma }-{\omega }_{\sigma \beta }^{\beta }\right)\right)\hfill \\ & +{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\sigma }^{\pi }\left((N+1)\left({\Gamma }_{\mu |\nu }-{\omega }_{\mu \alpha |\nu }^{\alpha }+{\omega }_{\mu \nu }^{\alpha }({\Gamma }_{\alpha }-{\omega }_{\alpha \beta }^{\beta })\right)+\left({\Gamma }_{\mu }-{\omega }_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\nu }-{\omega }_{\nu \beta }^{\beta }\right)\right),\hfill \end{array}$$

(20)

and the corresponding ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$.

3. Generalization of Geodesic Line and Geodesic Mapping

In this section, we will generalize the concept of $(\overline{m},m)$-conformal mappings [26] by adding a new tensor in the deformation tensor ${\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }$. This generalization aims to help physics to find more precise mathematical base of cosmological models.

Definition 1.

A curve $\mathsf{\ell }=\mathsf{\ell }\left(t\right)=\left({\mathsf{\ell }}^{\pi }\left(t\right)\right)$ in an N-dimensional Riemannian spacetime is called the m-conformal F-line if its coordinates ${\mathsf{\ell }}^{\pi }$ satisfy the next system of differential equations

$${\displaystyle \frac{d^2{\mathsf{\ell }}^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\mathsf{\ell }}^{\pi }}{dt}}+\left(m{g}_{\alpha \beta }{g}^{\pi \gamma }{\Gamma }_{\gamma }+2{\sigma }_{\alpha }{F}_{\beta }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}},$$

(21)

for time t, scalar functions ρ and m, a 1-form ${\sigma }_{\alpha }$, and affinor $F_{\alpha }^{\pi }$.

Remark 2.

If $m{g}_{\alpha \beta }{g}^{\pi \gamma }{\Gamma }_{\gamma }+2{\sigma }_{\alpha }{F}_{\beta }^{\pi }=0$, the m-conformal F-line reduces to a geodesic line of the spacetime ${\mathbb{R}}_N$. If $m=0$ and ${\sigma }_{\alpha }{F}_{\beta }^{\pi }\ne 0$, this curve becomes the F-plannar one.

Because ${\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}$ is symmetric by $\alpha$ and $\beta$, such as the metric tensor $g_{\alpha \beta }$, the definition of m-conformal F-line of spacetime ${\mathbb{R}}_N$ should be transformed to the curve ℓ whose components ${\mathsf{\ell }}^{\pi }$ satisfy the next symmetrized system of differential equations

$${\displaystyle \frac{d^2{\mathsf{\ell }}^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\mathsf{\ell }}^{\pi }}{dt}}+\left(m{g}_{\alpha \beta }{g}^{\pi \gamma }{\Gamma }_{\gamma }+{\sigma }_{\alpha }{F}_{\beta }^{\pi }+{\sigma }_{\beta }{F}_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}.$$

(22)

Definition 2.

A mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ which any $\overline{m}$-conformal $\overline{F}$-line of spacetime ${\overline{\mathbb{R}}}_N$ transforms to an m-conformal F-line of spacetime ${\mathbb{R}}_N$ is the $(\overline{m},m)$-conformal $(\overline{F},F)$ mapping.

Let us prove the next lemma.

Lemma 1.

A mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ is $(\overline{m},m)$-conformal $(\overline{F},F)$-mapping if and only if the Christoffel symbols ${\overline{\Gamma }}_{\mu \nu }^{\pi }$ and ${\Gamma }_{\mu \nu }^{\pi }$ of spacetimes ${\overline{\mathbb{R}}}_N$ and ${\mathbb{R}}_N$ satisfy the equation

$${\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }={\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }+m{g}_{\mu \nu }{g}^{\pi \gamma }{\Gamma }_{\gamma }+{\sigma }_{\nu }{F}_{\mu }^{\pi }+{\sigma }_{\mu }{F}_{\nu }^{\pi }-\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }-{\overline{\sigma }}_{\nu }{\overline{F}}_{\mu }^{\pi }-{\overline{\sigma }}_{\mu }{\overline{F}}_{\nu }^{\pi },$$

(23)

Proof. 

The systems of differential equations which determine the m-conformal F-line of spacetime ${\mathbb{R}}_N$ and the $\overline{m}$-conformal $\overline{F}$-line of space ${\overline{\mathbb{R}}}_N$ are

$$\left\{\begin{array}{c}{\displaystyle \frac{d^2{\mathsf{\ell }}^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\mathsf{\ell }}^{\pi }}{dt}}+\left(m{g}_{\alpha \beta }{g}^{\pi \gamma }{\Gamma }_{\gamma }+{\sigma }_{\alpha }{F}_{\beta }^{\pi }+{\sigma }_{\beta }{F}_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}},\hfill \\ {\displaystyle \frac{d^2{\mathsf{\ell }}^{\pi }}{d{t}^2}}+{\overline{\Gamma }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}=\overline{\rho }{\displaystyle \frac{d{\mathsf{\ell }}^{\pi }}{dt}}+\left(\overline{m}{\overline{g}}_{\alpha \beta }{\overline{g}}^{\pi \gamma }{\overline{\Gamma }}_{\gamma }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\beta }^{\pi }+{\overline{\sigma }}_{\beta }{\overline{F}}_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}.\hfill \end{array}\right.$$

(24)

The difference of these two equations gives

$$\begin{array}{cc}\hfill \left({\Gamma }_{\alpha \beta }^{\pi }-{\overline{\Gamma }}_{\alpha \beta }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}& =({\rho }_{\alpha }{\delta }_{\beta }^{\pi }+{\rho }_{\beta }{\delta }_{\alpha }^{\pi }){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}-\left({\overline{\rho }}_{\alpha }{\delta }_{\beta }^{\pi }+{\overline{\rho }}_{\beta }{\delta }_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}\hfill \\ & +\left(m{g}_{\alpha \beta }{g}^{\pi \gamma }{\Gamma }_{\gamma }+{\sigma }_{\alpha }{F}_{\beta }^{\pi }+{\sigma }_{\beta }{F}_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}}\hfill \\ & -\left(\overline{m}{\overline{g}}_{\alpha \beta }{\overline{g}}^{\pi \gamma }{\overline{\Gamma }}_{\gamma }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\beta }^{\pi }+{\overline{\sigma }}_{\beta }{\overline{F}}_{\alpha }^{\pi }\right){\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\mathsf{\ell }}^{\beta }}{dt}},\hfill \end{array}$$

(25)

for $\rho ={\rho }_{\alpha }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}$ and $\overline{\rho }={\overline{\rho }}_{\alpha }{\displaystyle \frac{d{\mathsf{\ell }}^{\alpha }}{dt}}$.

If ${\rho }_{\alpha }-{\overline{\rho }}_{\alpha }={\psi }_{\alpha }$, the equality (25) will be satisfied identically if and only if

$${\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }={\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }+m{g}_{\mu \nu }{g}^{\pi \gamma }{\Gamma }_{\gamma }+{\sigma }_{\nu }{F}_{\mu }^{\pi }+{\sigma }_{\mu }{F}_{\nu }^{\pi }-\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }-{\overline{\sigma }}_{\nu }{\overline{F}}_{\mu }^{\pi }-{\overline{\sigma }}_{\mu }{\overline{F}}_{\nu }^{\pi },$$

which completes the proof of this lemma. □

The difference ${\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }$ is a tensor of the type $(1,2)$. The geometrical objects ${\Gamma }_{\alpha }$ and ${\overline{\Gamma }}_{\alpha }$ are not tensors. Under transformation of coordinate system $Ox\to O^{\prime }{x}^{\prime }$, the geometrical objects $m{g}_{\mu \nu }{g}^{\pi \alpha }{\Gamma }_{\alpha }$ and $\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }$ transform as

$$\begin{array}{cc}\hfill m{g}_{{\mu }^{\prime }{\nu }^{\prime }}{g}^{{\pi }^{\prime }{\alpha }^{\prime }}{\Gamma }_{{\alpha }^{\prime }}& =m{g}_{\mu \nu }{g}^{\pi \alpha }{x}_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}{x}_{\alpha }^{{\alpha }^{\prime }}\left(x_{{\alpha }^{\prime }}^{\alpha }{\Gamma }_{\alpha }+x_{\beta }^{{\beta }^{\prime }}{x}_{{\alpha }^{\prime }{\beta }^{\prime }}^{\beta }\right)\hfill \\ & =x_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}m{g}_{\mu \nu }{g}^{\pi \alpha }{\Gamma }_{\alpha }+x_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}{x}_{\alpha }^{{\alpha }^{\prime }}{x}_{\beta }^{{\beta }^{\prime }}{x}_{{\alpha }^{\prime }{\beta }^{\prime }}^{\beta }m{g}_{\mu \nu }{g}^{\pi \alpha },\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \overline{m}{\overline{g}}_{{\mu }^{\prime }{\nu }^{\prime }}{\overline{g}}^{{\pi }^{\prime }{\alpha }^{\prime }}{\overline{\Gamma }}_{{\alpha }^{\prime }}& =\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{x}_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}{x}_{\alpha }^{{\alpha }^{\prime }}\left(x_{{\alpha }^{\prime }}^{\alpha }{\overline{\Gamma }}_{\alpha }+x_{\beta }^{{\beta }^{\prime }}{x}_{{\alpha }^{\prime }{\beta }^{\prime }}^{\beta }\right)\hfill \\ & =x_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }+x_{{\mu }^{\prime }}^{\mu }{x}_{{\nu }^{\prime }}^{\nu }{x}_{\pi }^{{\pi }^{\prime }}{x}_{\alpha }^{{\alpha }^{\prime }}{x}_{\beta }^{{\beta }^{\prime }}{x}_{{\alpha }^{\prime }{\beta }^{\prime }}^{\beta }\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha },\hfill \end{array}$$

(27)

The difference of relations (26) and (27) is a tensor if and only if $m{g}_{\mu \nu }{g}^{\pi \alpha }=\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }$. After composing this equality using $g^{\mu \nu }{g}_{\pi \alpha }$, we obtain $N^{2}m=\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{g}^{\mu \nu }{g}_{\pi \alpha }$, i.e.,

$$m={\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{g}^{\mu \nu }{g}_{\pi \alpha }.$$

(28)

The next lemma was proved in this way.

Lemma 2.

The scalar functions m and $\overline{m}$ in the basic Equation (23) satisfy the Equation (28). □

Remark 3.

In the case of a conformal mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$, defined as $g_{\alpha \beta }=e^{2\psi }{\overline{g}}_{\alpha \beta }$, for a scalar function ψ, the equality $g^{\alpha \beta }=e^{-2\psi }{\overline{g}}^{\alpha \beta }$ is satisfied, which induces

$$m={\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{e}^{-2\psi }{\overline{g}}^{\mu \nu }{e}^{2\psi }{\overline{g}}_{\pi \alpha }={\displaystyle \frac{1}{N^2}}\overline{m}{\delta }_{\mu }^{\mu }{\delta }_{\pi }^{\pi }=\overline{m},$$

(29)

i.e., $\delta m=m-\overline{m}=0$. Moreover, from the basic equation of conformal mapping analyzed in this remark,

$${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }-g_{\mu \nu }{g}^{\pi \alpha }{\psi }_{\alpha },$$

(30)

and its contraction by π and ν, ${\psi }_{\mu }={\displaystyle \frac{1}{N}}\left({\Gamma }_{\mu }-{\overline{\Gamma }}_{\mu }\right)$ was obtained. Because $g_{\mu \nu }{g}^{\pi \sigma }={\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \sigma }$ under conformal mappings (for details, see Equation (29)), we transform the basic Equation (30) to its equivalent form

$${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\psi }_{\nu }{\delta }_{\mu }^{\pi }+{\psi }_{\mu }{\delta }_{\nu }^{\pi }-{\displaystyle \frac{1}{N}}{g}_{\mu \nu }{g}^{\pi \alpha }{\Gamma }_{\alpha }+{\displaystyle \frac{1}{N}}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }.$$

(31)

With respect to the equality (31) and the basic Equation (23), we recognize $m=\overline{m}={\displaystyle \frac{1}{N}}$ and ${\sigma }_{\mu }{F}_{\nu }^{\pi }+{\sigma }_{\nu }{F}_{\mu }^{\pi }={\overline{\sigma }}_{\mu }{\overline{F}}_{\nu }^{\pi }+{\overline{\sigma }}_{\nu }{\overline{F}}_{\mu }^{\pi }$. The spatial case of the $(\overline{m},m)$-conformal $(\overline{F},F)$-mapping, the $(-{\displaystyle \frac{1}{N}},-{\displaystyle \frac{1}{N}})$-conformal $(0,0)$-mapping, is analyzed conformal mapping.

Motivated by the last remark, we will obtain a variation of scalar function m, $\delta m=m-\overline{m}$, in the next lemma.

Lemma 3.

In a cosmological model where $\delta g^{\mu \nu }\delta g_{\alpha \beta }$ is infinitesimal, the variation of coefficient of conformality m is neglected. If the product $\delta g^{\mu \nu }\delta g_{\alpha \beta }$ non-trivially impacts in variations, then $\delta m={\displaystyle \frac{1}{N^2}}\overline{m}\delta g^{\mu \nu }{g}_{\alpha \beta }{\overline{g}}_{\mu \nu }{\overline{g}}^{\alpha \beta }$.

Proof. 

The next equalities are satisfied

$$\begin{array}{cc}\hfill m& ={\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{g}^{\mu \nu }{g}_{\pi \alpha }\hfill \\ & ={\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }({\overline{g}}^{\mu \nu }+\delta g^{\mu \nu })({\overline{g}}_{\pi \alpha }+\delta g_{\pi \alpha })\hfill \\ & ={\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }\left({\overline{g}}^{\mu \nu }{\overline{g}}_{\pi \alpha }-{\overline{g}}^{\mu \nu }\delta g^{\gamma \delta }{\overline{g}}_{\pi \gamma }{\overline{g}}_{\alpha \delta }+\delta g^{\mu \nu }{\overline{g}}_{\pi \alpha }+\delta g^{\mu \nu }\delta g_{\pi \alpha }\right).\hfill \end{array}$$

The first three summands in the last equality are

$$\begin{gathered} {\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{g}}^{\mu \nu }{\overline{g}}_{\pi \alpha }=1, -{\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{g}}^{\mu \nu }\delta g^{\gamma \delta }{\overline{g}}_{\pi \gamma }{\overline{g}}_{\alpha \delta }=-{\displaystyle \frac{1}{N}}\overline{m}\delta g^{\gamma \delta }{\overline{g}}_{\gamma \delta }, \\ {\displaystyle \frac{1}{N^2}}\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }\delta g^{\mu \nu }{\overline{g}}_{\pi \alpha }={\displaystyle \frac{1}{N}}\overline{m}\delta g^{\mu \nu }{\overline{g}}_{\mu \nu }, \end{gathered}$$

(32)

In this way, and with respect to the corresponding exchange of indices in the second equality in (32), we get

$$m=\overline{m}(1+{\displaystyle \frac{1}{N^2}}\delta g^{\mu \nu }\delta g_{\pi \alpha }{\overline{g}}^{\mu \nu }{\overline{g}}_{\pi \alpha }).$$

which completes the proof that if $\left(\delta g^{\mu \nu }\delta g_{\pi \alpha }\right)\to 0$ then $m=\overline{m}$, but that variation of m has an impact in cosmological models if squared variance $\delta g_{\mu \nu }$ is not neglected. □

4. Invariants

In this section, we will apply the methodology from [12,13], which was reviewed in Section 2, to obtain invariants for an $(\overline{m},m)$-conformal $(\overline{F},F)$-mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$. After contracting the basic Equation (23) by $\pi$ and $\nu$, we obtain

$${\Gamma }_{\mu }={\overline{\Gamma }}_{\mu }+(N+1){\psi }_{\mu }+m{\Gamma }_{\mu }-\overline{m}{\overline{\Gamma }}_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F-{\overline{\sigma }}_{\alpha }{\overline{F}}_{\mu }^{\alpha }-{\overline{\sigma }}_{\mu }\overline{F},$$

i.e.,

$${\psi }_{\mu }=-{\displaystyle \frac{1}{N+1}}\left((m-1){\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F\right)+{\displaystyle \frac{1}{N+1}}\left((\overline{m}-1){\overline{\Gamma }}_{\mu }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\mu }^{\alpha }+{\overline{\sigma }}_{\mu }\overline{F}\right),$$

(33)

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

After substituting the expression (33) into the basic Equation (23), one transforms it to

$$\begin{array}{cc}\hfill {\Gamma }_{\mu \nu }^{\pi }& ={\overline{\Gamma }}_{\mu \nu }^{\pi }-{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left((m-1){\Gamma }_{\nu }+{\sigma }_{\alpha }{F}_{\nu }^{\alpha }+{\sigma }_{\nu }F\right)\hfill \\ & -{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left((m-1){\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F\right)+m{g}_{\mu \nu }{g}^{\pi \alpha }{\Gamma }_{\alpha }+{\sigma }_{\nu }{F}_{\mu }^{\pi }+{\sigma }_{\mu }{F}_{\nu }^{\pi }\hfill \\ & +{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left((\overline{m}-1){\overline{\Gamma }}_{\nu }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\nu }^{\alpha }+{\overline{\sigma }}_{\nu }\overline{F}\right)\hfill \\ & +{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left((\overline{m}-1){\overline{\Gamma }}_{\mu }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\mu }^{\alpha }+{\overline{\sigma }}_{\mu }\overline{F}\right)-\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }-{\overline{\sigma }}_{\nu }{\overline{F}}_{\mu }^{\pi }-{\overline{\sigma }}_{\mu }{\overline{F}}_{\nu }^{\pi }.\hfill \end{array}$$

(34)

To simplify writing, we involve the next exchanges

$$A_{\mu }=(m-1){\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F,$$

(35)

$${\overline{A}}_{\mu }=(\overline{m}-1){\overline{\Gamma }}_{\mu }+{\overline{\sigma }}_{\alpha }{\overline{F}}_{\mu }^{\alpha }+{\overline{\sigma }}_{\mu }\overline{F},$$

(36)

$$B_{\mu \nu }^{\pi }=m{g}_{\mu \nu }{g}^{\pi \alpha }{\Gamma }_{\alpha }+{\sigma }_{\nu }{F}_{\mu }^{\pi }+{\sigma }_{\mu }{F}_{\nu }^{\pi },$$

(37)

$${\overline{B}}_{\mu \nu }^{\pi }=\overline{m}{\overline{g}}_{\mu \nu }{\overline{g}}^{\pi \alpha }{\overline{\Gamma }}_{\alpha }+{\overline{\sigma }}_{\nu }{\overline{F}}_{\mu }^{\pi }+{\overline{\sigma }}_{\mu }{\overline{F}}_{\nu }^{\pi }.$$

(38)

The geometrical objects $B_{\mu \nu }^{\pi }$ and ${\overline{B}}_{\mu \nu }^{\pi }$ are symmetric by $\mu$ and $\nu$.

Proposition 1.

The geometrical objects $A_{\mu }$ and $B_{\mu \nu }^{\pi }$ defined by (35) and (37) satisfy the equation

$$A_{\mu }=B_{\mu \alpha }^{\alpha }-{\Gamma }_{\mu }.$$

(39)

Proof. 

Because

$$B_{\mu \alpha }^{\alpha }=m{g}_{\mu \alpha }{g}^{\alpha \gamma }{\Gamma }_{\gamma }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }{F}_{\alpha }^{\alpha }=m{\delta }_{\mu }^{\gamma }{\Gamma }_{\gamma }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F=m{\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F,$$

such as

$$A_{\mu }=(m-1){\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F=m{\Gamma }_{\mu }+{\sigma }_{\alpha }{F}_{\mu }^{\alpha }+{\sigma }_{\mu }F-{\Gamma }_{\mu }=B_{\mu \alpha }^{\alpha }-{\Gamma }_{\mu },$$

we conclude that the Equation (39) holds. □

With respect to the exchanges (35)–(38), and the relation (39), the basic Equation (34) transforms to

$$\begin{array}{cc}\hfill {\Gamma }_{\mu \nu }^{\pi }& ={\overline{\Gamma }}_{\mu \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu }\right)-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{B}_{\nu \alpha }^{\alpha }+{\delta }_{\nu }^{\pi }{B}_{\mu \alpha }^{\alpha }\right)+B_{\mu \nu }^{\pi }\hfill \\ & -{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\overline{\Gamma }}_{\nu }+{\delta }_{\nu }^{\pi }{\overline{\Gamma }}_{\mu }\right)-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\overline{B}}_{\nu \alpha }^{\alpha }+{\delta }_{\nu }^{\pi }{\overline{B}}_{\mu \alpha }^{\alpha }\right)-{\overline{B}}_{\mu \nu }^{\pi }.\hfill \end{array}$$

(40)

From the transformed basic Equation (40), we read

$${\theta }_{\mu \nu }^{\pi }={\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu }\right)-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{B}_{\nu \alpha }^{\alpha }+{\delta }_{\nu }^{\pi }{B}_{\mu \alpha }^{\alpha }\right)+B_{\mu \nu }^{\pi },$$

(41)

and the corresponding ${\overline{\theta }}_{\mu \nu }^{\pi }$.

That means that the basic equation of Thomas type for the mapping f is

$${\mathcal{T}}_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-B_{\mu \nu }^{\pi }-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu }\right)+{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{B}_{\nu \alpha }^{\alpha }+{\delta }_{\nu }^{\pi }{B}_{\mu \alpha }^{\alpha }\right).$$

(42)

The covariant derivative of ${\theta }_{\mu \nu }^{\pi }$ is

$${\theta }_{\mu \nu |\sigma }^{\pi }={\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{\Gamma }_{\nu |\sigma }+{\delta }_{\nu }^{\pi }{\Gamma }_{\mu |\sigma }\right)-{\displaystyle \frac{1}{N+1}}\left({\delta }_{\mu }^{\pi }{B}_{\nu \alpha |\sigma }^{\alpha }+{\delta }_{\nu }^{\pi }{B}_{\mu \alpha |\sigma }^{\alpha }\right)+B_{\mu \nu |\sigma }^{\pi }.$$

(43)

The antisymmetrization without division of (43) by $\nu$ and $\sigma$, together with the equality ${\Gamma }_{\mu |\nu }={\Gamma }_{\nu |\mu }$, gives

$$\begin{array}{cc}\hfill -{\theta }_{\mu \nu |\sigma }^{\pi }+{\theta }_{\mu \sigma |\nu }^{\pi }& ={\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }{B}_{\left[\nu \alpha \right|\sigma ]}^{\alpha }-B_{\mu \left[\nu \right|\sigma ]}^{\pi }\hfill \\ & -{\displaystyle \frac{1}{N+1}}{\delta }_{\nu }^{\pi }\left({\Gamma }_{\mu |\sigma }-B_{\mu \alpha |\sigma }^{\alpha }\right)+{\displaystyle \frac{1}{N+1}}{\delta }_{\sigma }^{\pi }\left({\Gamma }_{\mu |\nu }-B_{\mu \alpha |\nu }^{\alpha }\right).\hfill \end{array}$$

(44)

The composition ${\theta }_{\mu \nu }^{\alpha }{\theta }_{\alpha \sigma }^{\pi }$ antisymmetrized by $\nu$ and $\sigma$ is

$$\begin{array}{cc}\hfill {\theta }_{\mu \nu }^{\alpha }{\theta }_{\alpha \sigma }^{\pi }-{\theta }_{\mu \sigma }^{\alpha }{\theta }_{\alpha \nu }^{\pi }& =B_{\mu \nu }^{\alpha }{B}_{\alpha \sigma }^{\pi }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \nu }^{\pi }\hfill \\ & -{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\nu }^{\pi }\left((N+1)B_{\mu \sigma }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\sigma }-B_{\sigma \alpha }^{\alpha }\right)\right)\hfill \\ & +{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\sigma }^{\pi }\left((N+1)B_{\mu \nu }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\nu }-B_{\nu \alpha }^{\alpha }\right)\right).\hfill \end{array}$$

(45)

After summing the expressions (44) and (45) with curvature tensor $R_{\mu \nu \sigma }^{\pi }$, we obtain that the basic invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ of the mapping f is

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mu \nu \sigma }^{\pi }& =R_{\mu \nu \sigma }^{\pi }-B_{\mu \nu |\sigma }^{\pi }+B_{\mu \sigma |\nu }^{\pi }+B_{\mu \nu }^{\alpha }{B}_{\alpha \sigma }^{\pi }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left(B_{\nu \alpha |\sigma }^{\alpha }-B_{\sigma \alpha |\nu }^{\alpha }\right)\hfill \\ & -{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\nu }^{\pi }\left((N+1)\left({\Gamma }_{\mu |\sigma }-B_{\mu \alpha |\sigma }^{\alpha }+B_{\mu \sigma }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\sigma }-B_{\sigma \alpha }^{\alpha }\right)\right)\hfill \\ & +{\displaystyle \frac{1}{{(N+1)}^2}}{\delta }_{\sigma }^{\pi }\left((N+1)\left({\Gamma }_{\mu |\nu }-B_{\mu \alpha |\nu }^{\alpha }+B_{\mu \nu }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\nu }-B_{\nu \alpha }^{\alpha }\right)\right),\hfill \end{array}$$

(46)

and the corresponding ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$.

The trace ${\mathcal{W}}_{\mu \sigma }={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }$ of the invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ given by (46) is

$$\begin{array}{cc}\hfill {\mathcal{W}}_{\mu \sigma }& =R_{\mu \sigma }-B_{\mu \alpha |\sigma }^{\alpha }+B_{\mu \sigma |\alpha }^{\alpha }+B_{\mu \beta }^{\alpha }{B}_{\sigma \alpha }^{\beta }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \beta }^{\beta }+{\displaystyle \frac{1}{N+1}}\left(B_{\mu \alpha |\sigma }^{\alpha }-B_{\sigma \alpha |\mu }^{\alpha }\right)\hfill \\ & -{\displaystyle \frac{N-1}{{(N+1)}^2}}\left((N+1)\left({\Gamma }_{\mu |\sigma }-B_{\mu \alpha |\sigma }^{\alpha }+B_{\mu \sigma }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\sigma }-B_{\sigma \alpha }^{\alpha }\right)\right).\hfill \end{array}$$

(47)

The basic invariants ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ are written in the forms

$${\mathcal{W}}_{\mu \nu \sigma }^{\pi }=R_{\mu \nu \sigma }^{\pi }+Y_{\mu \nu \sigma }^{\pi }+{\delta }_{\mu }^{\pi }{X}_{\left[\nu \sigma \right]}+{\delta }_{\nu }^{\pi }{X}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{X}_{\mu \nu },$$

(48)

$${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }+{\overline{Y}}_{\mu \nu \sigma }^{\pi }+{\delta }_{\mu }^{\pi }{\overline{X}}_{\left[\nu \sigma \right]}+{\delta }_{\nu }^{\pi }{\overline{X}}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{\overline{X}}_{\mu \nu },$$

(49)

for

$$\begin{array}{cc}\hfill & X_{\mu \nu }=-{\displaystyle \frac{1}{{(N+1)}^2}}\left((N+1)\left({\Gamma }_{\mu |\nu }-B_{\mu \alpha |\nu }^{\alpha }+B_{\mu \nu }^{\alpha }\left({\Gamma }_{\alpha }-B_{\alpha \beta }^{\beta }\right)\right)+\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)({\Gamma }_{\nu }-B_{\nu \beta }^{\beta }\right),\hfill \\ \hfill & {\overline{X}}_{\mu \nu }=-{\displaystyle \frac{1}{{(N+1)}^2}}\left((N+1)\left({\overline{\Gamma }}_{\mu \parallel \nu }-{\overline{B}}_{\mu \alpha \parallel \nu }^{\alpha }+{\overline{B}}_{\mu \nu }^{\alpha }\left({\overline{\Gamma }}_{\alpha }-{\overline{B}}_{\alpha \beta }^{\beta }\right)\right)+\left({\overline{\Gamma }}_{\mu }-{\overline{B}}_{\mu \alpha }^{\alpha }\right)({\overline{\Gamma }}_{\nu }-{\overline{B}}_{\nu \beta }^{\beta }\right),\hfill \\ \hfill & Y_{\mu \nu \sigma }^{\pi }=-B_{\mu \nu |\sigma }^{\pi }+B_{\mu \sigma |\nu }^{\pi }+B_{\mu \nu }^{\alpha }{B}_{\sigma \alpha }^{\pi }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \nu }^{\pi },\hfill \\ \hfill & {\overline{Y}}_{\mu \nu \sigma }^{\pi }=-{\overline{B}}_{\mu \nu \parallel \sigma }^{\pi }+{\overline{B}}_{\mu \sigma \parallel \nu }^{\pi }+{\overline{B}}_{\mu \nu }^{\alpha }{\overline{B}}_{\sigma \alpha }^{\pi }-{\overline{B}}_{\mu \sigma }^{\alpha }{\overline{B}}_{\alpha \nu }^{\pi }.\hfill \end{array}$$

The differences $0={\mathcal{W}}_{\alpha \nu \sigma }^{\alpha }-{\overline{\mathcal{W}}}_{\alpha \nu \sigma }^{\alpha }$ and $0={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }-{\overline{\mathcal{W}}}_{\mu \alpha \sigma }^{\alpha }$ are equivalent to the next system of equations

$$S:\left\{\begin{array}{c}\left(Y_{\alpha \nu \sigma }^{\alpha }-{\overline{Y}}_{\alpha \nu \sigma }^{\alpha }\right)+(N+1)\left(X_{\left[\nu \sigma \right]}-{\overline{X}}_{\left[\nu \sigma \right]}\right)=0,\hfill \\ \left(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }\right)+\left(Y_{\mu \alpha \sigma }^{\alpha }-{\overline{Y}}_{\mu \alpha \sigma }^{\alpha }\right)+\left(X_{\left[\mu \sigma \right]}-{\overline{X}}_{\left[\mu \sigma \right]}\right)+(N-1)\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right)=0.\hfill \end{array}\right.$$

(50)

From the first equation of system S, we obtain

$$\left(X_{\left[\mu \sigma \right]}-{\overline{X}}_{\left[\mu \sigma \right]}\right)=-{\displaystyle \frac{1}{N+1}}\left(Y_{\alpha \nu \sigma }^{\alpha }-{\overline{Y}}_{\alpha \nu \sigma }^{\alpha }\right).$$

(51)

After substituting the expression (51) into the second equation of system S, we get

$$\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right)=-{\displaystyle \frac{1}{N-1}}\left(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }\right)-{\displaystyle \frac{1}{N-1}}\left(Y_{\mu \alpha \sigma }^{\alpha }-{\overline{Y}}_{\mu \alpha \sigma }^{\alpha }\right)+{\displaystyle \frac{1}{N^2-1}}\left(Y_{\alpha \mu \sigma }^{\alpha }-{\overline{Y}}_{\alpha \mu \sigma }^{\alpha }\right).$$

(52)

After involving the solutions (51) and (52) of the system S into the equality $0={\mathcal{W}}_{\mu \nu \sigma }^{\pi }-{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$, i.e., into the relation

$$\begin{array}{cc}\hfill 0& =\left(R_{\mu \nu \sigma }^{\pi }-{\overline{R}}_{\mu \nu \sigma }^{\pi }\right)\hfill \\ & +\left(Y_{\mu \nu \sigma }^{\pi }-{\overline{Y}}_{\mu \nu \sigma }^{\pi }\right)+{\delta }_{\mu }^{\pi }\left(X_{\left[\nu \sigma \right]}-{\overline{X}}_{\left[\nu \sigma \right]}\right)+{\delta }_{\nu }^{\pi }\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right)-{\delta }_{\sigma }^{\pi }\left(X_{\mu \nu }-{\overline{X}}_{\mu \nu }\right),\hfill \end{array}$$

we obtain the next equality

$$\begin{array}{cc}\hfill 0& =\left(R_{\mu \nu \sigma }^{\pi }-{\overline{R}}_{\mu \nu \sigma }^{\pi }\right)+\left(Y_{\mu \nu \sigma }^{\pi }-{\overline{Y}}_{\mu \nu \sigma }^{\pi }\right)-{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left(Y_{\alpha \nu \sigma }^{\alpha }-{\overline{Y}}_{\alpha \nu \sigma }^{\alpha }\right)\hfill \\ & -{\displaystyle \frac{1}{N-1}}{\delta }_{\nu }^{\pi }\left(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }\right)-{\displaystyle \frac{1}{N-1}}{\delta }_{\nu }^{\pi }\left(Y_{\mu \alpha \sigma }^{\alpha }-{\overline{Y}}_{\mu \alpha \sigma }^{\alpha }\right)+{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left(Y_{\alpha \mu \sigma }^{\alpha }-{\overline{Y}}_{\alpha \mu \sigma }^{\alpha }\right)\hfill \\ & +{\displaystyle \frac{1}{N-1}}{\delta }_{\sigma }^{\pi }\left(R_{\mu \nu }-{\overline{R}}_{\mu \nu }\right)+{\displaystyle \frac{1}{N-1}}{\delta }_{\sigma }^{\pi }\left(Y_{\mu \alpha \nu }^{\alpha }-{\overline{Y}}_{\mu \alpha \nu }^{\alpha }\right)-{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left(Y_{\alpha \mu \nu }^{\alpha }-{\overline{Y}}_{\alpha \mu \nu }^{\alpha }\right).\hfill \end{array}$$

(53)

This equality is equivalent to $W_{\mu \nu \sigma }^{\pi }={\overline{W}}_{\mu \nu \sigma }^{\pi }$, for

$$\begin{array}{cc}\hfill W_{\mu \nu \sigma }^{\pi }& =R_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{R}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\right)+Y_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }{Y}_{\alpha \nu \sigma }^{\alpha }\hfill \\ & -{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left((N+1)Y_{\mu \alpha \sigma }^{\alpha }-Y_{\alpha \mu \sigma }^{\alpha }\right)+{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left((N+1)Y_{\mu \alpha \nu }^{\alpha }-Y_{\alpha \mu \nu }^{\alpha }\right).\hfill \end{array}$$

(54)

After substituting the backward relations $Y_{\alpha \mu \nu }^{\alpha }=-B_{\mu \alpha |\nu }^{\alpha }+B_{\nu \alpha |\mu }^{\alpha }$ and $Y_{\mu \alpha \nu }^{\alpha }=-B_{\mu \alpha |\nu }^{\alpha }+B_{\mu \nu |\alpha }^{\alpha }+B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-B_{\mu \nu }^{\alpha }{B}_{\alpha \beta }^{\beta }$, such as the definition of $Y_{\mu \nu \sigma }^{\pi }$, into the expression (54), we transform it to

$$\begin{array}{cc}\hfill W_{\mu \nu \sigma }^{\pi }& =R_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{R}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\right)\hfill \\ & -B_{\mu \nu |\sigma }^{\pi }+B_{\mu \sigma |\nu }^{\pi }+B_{\mu \nu }^{\alpha }{B}_{\alpha \sigma }^{\pi }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left(B_{\nu \alpha |\sigma }^{\alpha }-B_{\sigma \alpha |\nu }^{\alpha }\right)\hfill \\ & -{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left(N{B}_{\mu \alpha |\sigma }^{\alpha }+B_{\sigma \alpha |\mu }^{\alpha }-(N+1)B_{\mu \sigma |\alpha }^{\alpha }-(N+1)\left(B_{\mu \beta }^{\alpha }{B}_{\sigma \alpha }^{\beta }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \beta }^{\beta }\right)\right)\hfill \\ & +{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left(N{B}_{\mu \alpha |\nu }^{\alpha }+B_{\nu \alpha |\mu }^{\alpha }-(N+1)B_{\nu \sigma |\alpha }^{\alpha }-(N+1)\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-B_{\mu \nu }^{\alpha }{B}_{\alpha \beta }^{\beta }\right)\right).\hfill \end{array}$$

(55)

The traces $W_{\alpha \nu \sigma }^{\alpha }$, $W_{\mu \alpha \sigma }^{\alpha }$, and $W_{\mu \nu \alpha }^{\alpha }$ of the geometrical object $W_{\mu \nu \sigma }^{\pi }$ given by (55) are

$$\begin{array}{ccc}{W}_{\alpha \nu \sigma }^{\alpha }=0,& W_{\mu \alpha \sigma }^{\alpha }=0,& W_{\mu \nu \alpha }^{\alpha }=0.\end{array}$$

(56)

The next theorem was proved above.

Theorem 1.

Let $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ be an $(\overline{m},m)$-conformal $(\overline{F},F)$-planar mapping. The geometrical object ${\mathcal{T}}_{\mu \nu }^{\pi }$ given by (42) is the basic invariant of Thomas type for the mapping f. The geometrical object ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ given by (46) is the basic invariant of Weyl type for the mapping f. The geometrical object $W_{\mu \nu \sigma }^{\pi }$ given by (55) is the derived invariant of Weyl type for the mapping f.

The invariants ${\mathcal{T}}_{\mu \nu }^{\pi }$ and $W_{\mu \nu \sigma }^{\pi }$ are trace-free. The trace ${\mathcal{W}}_{\mu \sigma }={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }$ of the basic invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ is given by (47). □

Remark 4.

With respect to the classical methodology presented in [3], one invariant of Thomas type and one invariant of Weyl type are obtained for a mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$. These classical invariants actually correspond to what we define as the second class (derived invariants). In [12], this methodology was further developed by discovering a more fundamental class of invariants that chronologically and structurally precede the classical ones. For this reason, the invariants introduced in [12] are termed basic invariants of Weyl type, whereas the classical ones obtained by Weyl are referred to as derived invariants.

5. EXAMPLES: Action, Variation, and Two Compositions

The Weyl conformal tensor is inherently trace-free and is traditionally employed to derive gravitoelectric and gravitomagnetic tensors through orthogonal decomposition [28]. Within the classical framework of Weyl’s methodology [3], no invariant exists for a geometric mapping that yields a non-vanishing trace equivalent to the Ricci tensor. Motivated by this restrictive nature of classical invariants, and building upon the results presented in [12,13], the present research is undertaken to address these theoretical gaps.

We will analyze the generalized Einstein–Hilbert action in an N-dimensional space

$$2\kappa S_{mF}^{+}=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{\mathcal{W}}_{\alpha \beta },$$

(57)

for the trace ${\mathcal{W}}_{\alpha \beta }$ of invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ for the $(\overline{m},m)$-conformal $(\overline{F},F)$-mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$.

After composing the geometrical object ${\mathcal{W}}_{\mu \sigma }$ given by (47) with $g^{\mu \sigma }$, we get

$$\begin{array}{cc}\hfill g^{\alpha \beta }{\mathcal{W}}_{\alpha \beta }& =g^{\alpha \beta }{R}_{\alpha \beta }-{\displaystyle \frac{2}{N+1}}{\left(g^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }\right)}_{|\beta }+{\left(g^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }\right)}_{|\gamma }-{\displaystyle \frac{N-1}{N+1}}{\left(g^{\alpha \beta }{\Gamma }_{\alpha }\right)}_{|\beta }\hfill \\ & +g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }-{\displaystyle \frac{N-1}{N+1}}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }-{\displaystyle \frac{2}{N+1}}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{B}_{\gamma \delta }^{\delta }-{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }\hfill \\ & +{\displaystyle \frac{2(N-1)}{{(N+1)}^2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \gamma }^{\gamma }-{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }.\hfill \end{array}$$

(58)

In this model, the geometrical object $F_{\beta }^{\alpha }$ is $F_{\beta }^{\alpha }=g^{\alpha \gamma }{F}_{\gamma \beta }$, for a tensor $F_{\gamma \beta }$ of the type $(0,2)$, such that $\delta F_{\gamma \beta }=0$. Because the tensor $F_{\gamma \beta }$ is non-symmetric in general, its symmetric and anti-symmetric parts are

$$\begin{array}{cc}{F}_{\underline{\gamma \beta }}={\displaystyle \frac{1}{2}}\left(F_{\gamma \beta }+F_{\beta \gamma }\right),& F_{\underset{\vee }{\gamma \beta }}={\displaystyle \frac{1}{2}}\left(F_{\gamma \beta }-F_{\beta \gamma }\right).\end{array}$$

(59)

Hence, $F_{\gamma \beta }=F_{\underline{\gamma \beta }}+F_{\underset{\vee }{\gamma \beta }}$ and $F=g^{\gamma \beta }{F}_{\gamma \beta }=g^{\gamma \beta }{F}_{\underline{\gamma \beta }}$.

The 1-forms ${\overline{\sigma }}_{\alpha }$ and ${\sigma }_{\alpha }$ in the definition of $B_{\mu \nu }^{\pi }$ given by (37) and (38) are ${\sigma }_{\alpha }={\sigma }_{|\alpha }$ and ${\overline{\sigma }}_{\alpha }={\sigma }_{\parallel \alpha }$, where $\sigma$ and $\overline{\sigma }$ are scalar functions. For simplicity, and without limiting future research to this specific case, we also assume that $\delta \sigma =\sigma -\overline{\sigma }=0$.

With respect to the variation $\delta \left({\delta }_{\mu }^{\pi }\right)=\delta \left(g^{\pi \alpha }{g}_{\mu \alpha }\right)=\delta g^{\pi \alpha }{g}_{\mu \alpha }+g^{\pi \alpha }\delta g_{\mu \alpha }$, we obtain

$$\delta g_{\alpha \beta }=-\delta g^{\mu \nu }{g}_{\mu \alpha }{g}_{\nu \beta }.$$

(60)

Proposition 2.

The variation of ${\Gamma }_{\alpha }$ is

$$\delta {\Gamma }_{\alpha }=-{\displaystyle \frac{1}{2}}{\left(\delta g^{\mu \nu }{g}_{\mu \nu }\right)}_{,\alpha }=-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\left(\delta g^{\mu \nu }\right)}_{,\alpha }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu ,\alpha }\delta g^{\mu \nu }.$$

(61)

Proof. 

The variation of ${\Gamma }_{\alpha }={\displaystyle \frac{1}{2}}{g}^{\gamma \delta }{g}_{\gamma \delta ,\alpha }$ is

$$\begin{array}{cc}\hfill \delta {\Gamma }_{\alpha }& ={\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }+{\displaystyle \frac{1}{2}}{g}^{\gamma \delta }{\left(\delta g_{\gamma \delta }\right)}_{,\alpha }\hfill \\ & ={\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }+{\displaystyle \frac{1}{2}}{g}^{\gamma \delta }{\left(-\delta g^{\mu \nu }{g}_{\mu \gamma }{g}_{\nu \delta }\right)}_{,\alpha }\hfill \\ & ={\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }-{\displaystyle \frac{1}{2}}{\left(\delta g^{\mu \nu }\right)}_{,\alpha }{g}^{\gamma \delta }{g}_{\mu \gamma }{g}_{\nu \delta }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}^{\gamma \delta }{g}_{\mu \gamma ,\alpha }{g}_{\nu \delta }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}^{\gamma \delta }{g}_{\mu \gamma }{g}_{\nu \delta ,\alpha }\hfill \\ & ={\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }-{\displaystyle \frac{1}{2}}{\left(\delta g^{\mu \nu }\right)}_{,\alpha }{\delta }_{\mu }^{\delta }{g}_{\nu \delta }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{\delta }_{\nu }^{\gamma }{g}_{\mu \gamma ,\alpha }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{\delta }_{\mu }^{\delta }{g}_{\nu \delta ,\alpha }\hfill \\ & ={\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }-{\displaystyle \frac{1}{2}}{\left(\delta g^{\mu \nu }\right)}_{,\alpha }{g}_{\nu \mu }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\nu \mu ,\alpha }\hfill \\ & =-{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu ,\alpha }-{\displaystyle \frac{1}{2}}{\left(\delta g^{\mu \nu }\right)}_{,\alpha }{g}_{\mu \nu },\hfill \end{array}$$

which completes the proof of this proposition. □

From Lemma 3, and with respect to the model where $\delta g^{\mu \nu }$ linearly impacts to variations, we get

$$\delta m=0.$$

(62)

The next relation is also satisfied

$$\delta \left({\sigma }_{\beta }{F}_{\gamma }^{\mu }+{\sigma }_{\gamma }{F}_{\beta }^{\mu }\right)=\delta (g^{\mu \nu }\left({\sigma }_{\beta }{F}_{\nu \gamma }+{\sigma }_{\gamma }{F}_{\nu \beta }\right)=\delta g^{\mu \nu }\left({\sigma }_{\beta }{F}_{\nu \gamma }+{\sigma }_{\gamma }{F}_{\nu \beta }\right).$$

(63)

Proposition 3.

The variation of $B_{\beta \gamma }^{\alpha }$ is

$$\delta B_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{ϵ}_{\beta \gamma \mu \nu }^{\alpha \delta }.$$

(64)

The trace of this variation is

$$\delta B_{\beta \alpha }^{\alpha }=\delta g^{\mu \nu }{\tilde{\mathcal{B}}}_{\beta \alpha \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\beta }{\widetilde{ϵ}}_{\mu \nu }.$$

(65)

The geometrical object ${\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\alpha }$ and ${\widetilde{ϵ}}_{\mu \nu }$ are

$$\begin{array}{cc}\hfill {\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\alpha }& =-m{g}_{\mu \beta }{g}_{\nu \gamma }{g}^{\alpha \delta }{\Gamma }_{\delta }+m{g}_{\beta \gamma }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }{\Gamma }_{\delta }+{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }({\sigma }_{\beta }{F}_{\delta \gamma }+{\sigma }_{\gamma }{F}_{\delta \beta })\hfill \\ & +{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\left(m{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }-{\displaystyle \frac{1}{2}}{\left(m{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta },\hfill \end{array}$$

(66)

$${\tilde{\epsilon }}_{\beta \gamma \mu \nu }^{\alpha \delta }=-{\displaystyle \frac{1}{2}}m{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta },{\tilde{\epsilon }}_{\mu \nu }=-{\displaystyle \frac{1}{2}}m{g}_{\mu \nu }.$$

(67)

Proof. 

With respect to the Equations (61)–(63), we conclude that the variation of $B_{\mu \nu }^{\pi }$ given by (37) satisfies the next relations

$$\begin{array}{cc}\hfill \delta B_{\beta \gamma }^{\alpha }& =m\delta g_{\beta \gamma }{g}^{\alpha \delta }{\Gamma }_{\delta }+m{g}_{\beta \gamma }\delta g^{\alpha \delta }{\Gamma }_{\delta }+m{g}_{\beta \gamma }{g}^{\alpha \delta }\delta {\Gamma }_{\delta }+\delta g^{\alpha \delta }\left({\sigma }_{\beta }{F}_{\delta \gamma }+{\sigma }_{\gamma }{F}_{\delta \beta }\right)\hfill \\ & =-m\delta g^{\mu \nu }{g}_{\beta \mu }{g}_{\gamma \nu }{g}^{\alpha \delta }{\Gamma }_{\delta }+m\delta g^{\mu \nu }{g}_{\beta \gamma }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }{\Gamma }_{\delta }-{\displaystyle \frac{1}{2}}m{g}_{\beta \gamma }{g}^{\alpha \delta }{\left(\delta g^{\mu \nu }{g}_{\mu \nu }\right)}_{,\delta }\hfill \\ & +\delta g^{\mu \nu }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }\left({\sigma }_{\beta }{F}_{\delta \gamma }+{\sigma }_{\gamma }{F}_{\delta \beta }\right)\hfill \\ & =-m\delta g^{\mu \nu }{g}_{\beta \mu }{g}_{\gamma \nu }{g}^{\alpha \delta }{\Gamma }_{\delta }+m\delta g^{\mu \nu }{g}_{\beta \gamma }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }{\Gamma }_{\delta }-{\displaystyle \frac{1}{2}}{\left(m{g}_{\beta \gamma }{g}^{\alpha \delta }\delta g^{\mu \nu }{g}_{\mu \nu }\right)}_{,\delta }\hfill \\ & +{\displaystyle \frac{1}{2}}\delta g^{\mu \nu }{g}_{\mu \nu }{\left(m{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }+\delta g^{\mu \nu }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }\left({\sigma }_{\beta }{F}_{\delta \gamma }+{\sigma }_{\gamma }{F}_{\delta \beta }\right)\hfill \end{array}$$

Based on these equalities, we get

$$\delta B_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{\mathcal{B}}_{\beta \gamma \mu \nu }^{\alpha }-{\displaystyle \frac{1}{2}}{\left(m\delta g^{\mu \nu }{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta },$$

(68)

where ${\mathcal{B}}_{\beta \gamma \mu \nu }^{\alpha }=-m{g}_{\mu \beta }{g}_{\nu \gamma }{g}^{\alpha \delta }{\Gamma }_{\delta }+m{g}_{\beta \gamma }{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }{\Gamma }_{\delta }+{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{\left(m{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }+{\delta }_{\mu }^{\alpha }{\delta }_{\nu }^{\delta }({\sigma }_{\beta }{F}_{\delta \gamma }+{\sigma }_{\gamma }{F}_{\delta \beta })$.

After expressing ${\left(m\delta g^{\mu \nu }{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }$ as

$${\left(m\delta g^{\mu \nu }{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }=\delta g^{\mu \nu }{\left(m{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }\left(m{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right),$$

we obtain ${\left(m\delta g^{\mu \nu }{g}_{\mu \nu }{g}_{\beta \gamma }{g}^{\alpha \delta }\right)}_{,\delta }=\delta g^{\mu \nu }{ϵ}_{\beta \gamma \mu \nu ,\delta }^{\alpha \delta }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{ϵ}_{\beta \gamma \mu \nu }^{\alpha \delta }$, where ${\epsilon }_{\beta \gamma \mu \nu }^{\alpha \delta }$ is given by (67) Hence, the relation (68) transforms to

$$\delta B_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{\epsilon }_{\beta \gamma \mu \nu }^{\alpha \delta }.$$

The trace of variation (64) by $\alpha$ and $\gamma$ is

$$\delta B_{\beta \alpha }^{\alpha }=\delta g^{\mu \nu }{\tilde{\mathcal{B}}}_{\beta \alpha \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\beta }{\tilde{\epsilon }}_{\mu \nu }.$$

where we used ${\epsilon }_{\beta \alpha \mu \nu }^{\alpha \delta }=-{\displaystyle \frac{1}{2}}{\delta }_{\beta }^{\delta }m{g}_{\mu \nu }={\delta }_{\beta }^{\delta }{\tilde{\epsilon }}_{\mu \nu }$ and composed it with ${\left(\delta g^{\mu \nu }\right)}_{,\delta }$, which completed the proof of this proposition. □

Based on the relation (58), we transform the action (57) to

$$\begin{array}{cc}\hfill 2\kappa S_{mF}^{+}& =\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{R}_{\alpha \beta }-{\displaystyle \frac{2}{N+1}}\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }\right)}_{|\beta }\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }\right)}_{|\gamma }-{\displaystyle \frac{N-1}{N+1}}\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{\Gamma }_{\alpha }\right)}_{|\beta }\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }-{\displaystyle \frac{N-1}{N+1}}\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }\hfill \\ & -{\displaystyle \frac{2}{N+1}}\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{B}_{\gamma \delta }^{\delta }-{\displaystyle \frac{N-1}{{(N+1)}^2}}\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }\hfill \\ & +{\displaystyle \frac{2(N-1)}{{(N+1)}^2}}\int d^{N}x\sqrt{-g}{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \gamma }^{\gamma }-{\displaystyle \frac{N-1}{{(N+1)}^2}}\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }.\hfill \end{array}$$

(69)

The expression (69) may be represented in the form

$$\begin{array}{cc}\hfill 2\kappa S_{mF}^{+}& =I_1-{\displaystyle \frac{2}{N+1}}{I}_2+I_3-{\displaystyle \frac{N-1}{N+1}}{I}_4+I_5-{\displaystyle \frac{N-1}{N+1}}{I}_6\hfill \\ & -{\displaystyle \frac{2}{N+1}}{I}_7-{\displaystyle \frac{N-1}{{(N+1)}^2}}{I}_8+{\displaystyle \frac{2(N-1)}{{(N+1)}^2}}{I}_9-{\displaystyle \frac{N-1}{{(N+1)}^2}}{I}_{10},\hfill \end{array}$$

(70)

where

$$\begin{gathered} I_1=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{R}_{\alpha \beta }, I_2=\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }\right)}_{|\beta }{I}_3=\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }\right)}_{|\gamma }, \\ {I}_4=\int d^{N}x\sqrt{\left|g\right|}{\left(g^{\alpha \beta }{\Gamma }_{\alpha }\right)}_{|\beta }{I}_5=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }, I_6=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }, \\ {I}_7=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{B}_{\gamma \delta }^{\delta }, I_8=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }, \\ {I}_9=\int d^{N}x\sqrt{-g}{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \gamma }^{\gamma }, I_{10}=\int d^{N}x\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }. \end{gathered}$$

Proposition 4.

If ${\tau }^{\alpha }$ is a contravariant vector, the product $\sqrt{\left|g\right|}{\tau }_{|\alpha }^{\alpha }$ is a total differential.

Proof. 

The next equality is satisfied

$$\sqrt{\left|g\right|}{\tau }_{|\alpha }^{\alpha }=\sqrt{\left|g\right|}{\tau }_{,\alpha }^{\alpha }+\sqrt{\left|g\right|}{\Gamma }_{\beta \alpha }^{\alpha }{\tau }^{\beta }.$$

Because ${\Gamma }_{\alpha }={\displaystyle \frac{\partial \left|g\right|/\partial x^{\alpha }}{\left|g\right|}}$, i.e., $\partial \left|g\right|/\partial x^{\alpha }={\Gamma }_{\alpha }\sqrt{\left|g\right|}$, we transform the expression $\sqrt{\left|g\right|}{\tau }_{|\alpha }^{\alpha }$ to

$$\sqrt{\left|g\right|}{\tau }_{|\alpha }^{\alpha }=\sqrt{\left|g\right|}{\tau }_{,\alpha }^{\alpha }+(\partial \sqrt{\left|g\right|}/\partial x^{\alpha }){\tau }^{\alpha }={\left(\sqrt{\left|g\right|}{\tau }^{\alpha }\right)}_{,\alpha },$$

which completes the proof of this proposition. □

Let us prove the next Theorem.

Theorem 2.

Let $X_{\beta \gamma }^{\alpha }$ and $Y_{\beta \gamma }^{\alpha }$ be geometrical objects of types $(1,2)$ whose variations are $\delta X_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{U}_{\beta \gamma \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \delta }$ and $\delta Y_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{V}_{\beta \gamma \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{\tilde{V}}_{\beta \gamma \mu \nu }^{\alpha \delta }$. The variation of geometrical object $\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }$ is

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\right)& =-{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+\sqrt{\left|g\right|}\delta g^{\mu \nu }{\delta }_{\mu }^{\pi }{\delta }_{\nu }^{\sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left(U_{\beta \gamma \mu \nu }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{V}_{ϵ\zeta \mu \nu }^{\delta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }{\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)}_{,\theta }\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(g^{\pi \sigma }{\Gamma }_{\theta }+g_{,\theta }^{\pi \sigma }\right)\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\hfill \\ & \hfill \\ & +{\left\{\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\right\}}_{,\theta }.\hfill \end{array}$$

(71)

Proof. 

The variation of $X_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }$ is

$$\begin{array}{cc}\hfill \delta \left(X_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\right)& =\delta X_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }\delta Y_{ϵ\zeta }^{\delta }\hfill \\ & =\delta g^{\mu \nu }\left(U_{\beta \gamma \mu \nu }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{V}_{ϵ\zeta \mu \nu }^{\delta }\right)+{\left(\delta g^{\mu \nu }\right)}_{,\theta }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right).\hfill \end{array}$$

(72)

With respect to the result (72), we obtain that the variation of $\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }$ is

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\right)& =\delta \sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+\sqrt{\left|g\right|}\delta g^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+\sqrt{\left|g\right|}{g}^{\pi \sigma }\delta \left(X_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\right)\hfill \\ & =-{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+\sqrt{\left|g\right|}\delta g^{\mu \nu }{\delta }_{\mu }^{\pi }{\delta }_{\nu }^{\sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left(U_{\beta \gamma \mu \nu }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{V}_{ϵ\zeta \mu \nu }^{\delta }\right)\hfill \\ & +\sqrt{\left|g\right|}{\left(\delta g^{\mu \nu }\right)}_{,\theta }{g}^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right).\hfill \end{array}$$

(73)

Because $\partial \sqrt{\left|g\right|}/\partial x^{\alpha }=\sqrt{\left|g\right|}{\Gamma }_{\alpha }$, the last summand in the Equation (73) takes the form

$$\begin{array}{cc}\hfill \sqrt{\left|g\right|}{\left(\delta g^{\mu \nu }\right)}_{,\theta }{g}^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)& ={\left\{\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\right\}}_{,\theta }\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }{\Gamma }_{\theta }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{,\theta }^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)}_{,\theta }.\hfill \end{array}$$

(74)

After substituting the equality (74) in the relation (73), we obtain

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\right)& =-{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+\sqrt{\left|g\right|}\delta g^{\mu \nu }{\delta }_{\mu }^{\pi }{\delta }_{\nu }^{\sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{ϵ\zeta }^{\delta }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left(U_{\beta \gamma \mu \nu }^{\alpha }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{V}_{ϵ\zeta \mu \nu }^{\delta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(g^{\pi \sigma }{\Gamma }_{\theta }+g_{,\theta }^{\pi \sigma }\right)\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }{\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)}_{,\theta }\hfill \\ & +{\left\{\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left({\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{ϵ\zeta }^{\delta }+X_{\beta \gamma }^{\alpha }{\tilde{V}}_{ϵ\zeta \mu \nu }^{\delta \theta }\right)\right\}}_{,\theta },\hfill \end{array}$$

which completes the proof of this theorem. □

Corollary 1.

Let $X_{\beta \gamma }^{\alpha }$ and $Y_{\alpha }$ be geometrical objects of types $(1,2)$ and $(0,1)$, respectively, whose variations are $\delta X_{\beta \gamma }^{\alpha }=\delta g^{\mu \nu }{U}_{\beta \gamma \mu \nu }^{\alpha }+{\left(\delta g^{\mu \nu }\right)}_{,\delta }{\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \delta }$ and $\delta Y_{\alpha }=\delta g^{\mu \nu }{V}_{\alpha \mu \nu }+{\left(\delta g^{\mu \nu }\right)}_{,\alpha }{\tilde{V}}_{\mu \nu }$. The variation of geometrical object $\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{\delta }$ is

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{\delta }\right)& =-{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{\delta }+\sqrt{\left|g\right|}\delta g^{\mu \nu }{\delta }_{\mu }^{\pi }{\delta }_{\nu }^{\sigma }{X}_{\beta \gamma }^{\alpha }{Y}_{\delta }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\pi \sigma }\left(U_{\beta \gamma \mu \nu }^{\alpha }{Y}_{\delta }+X_{\beta \gamma }^{\alpha }{V}_{\delta \mu \nu }-{\Gamma }_{\theta }{\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{\delta }-{\Gamma }_{\delta }{X}_{\beta \gamma }^{\alpha }{\tilde{V}}_{\mu \nu }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }\left({\left(g^{\pi \sigma }{\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{\delta }\right)}_{,\theta }+{\left(g^{\pi \sigma }{X}_{\beta \gamma }^{\alpha }{\tilde{V}}_{\mu \nu }\right)}_{,\delta }\right)\hfill \\ & +{\left(\sqrt{\left|g\right|}{g}^{\pi \sigma }\delta g^{\mu \nu }{\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }{Y}_{\delta }\right)}_{,\theta }+{\left(\sqrt{\left|g\right|}{g}^{\pi \sigma }\delta g^{\mu \nu }{X}_{\beta \gamma }^{\alpha }{\tilde{V}}_{\mu \nu }\right)}_{,\delta }.\hfill \end{array}$$

(75)

The variation of the Ricci tensor $R_{\alpha \beta }=R_{\alpha \gamma \beta }^{\gamma }$ is

$$\begin{array}{cc}\hfill \delta R_{\alpha \beta }& ={\left(\delta {\Gamma }_{\alpha \gamma }^{\gamma }\right)}_{,\beta }-{\left(\delta {\Gamma }_{\alpha \beta }^{\gamma }\right)}_{,\gamma }+\delta {\Gamma }_{\alpha \gamma }^{\delta }{\Gamma }_{\beta \delta }^{\gamma }+{\Gamma }_{\alpha \gamma }^{\delta }\delta {\Gamma }_{\delta \gamma }^{\gamma }-\delta {\Gamma }_{\alpha \beta }^{\delta }{\Gamma }_{\delta \gamma }^{\gamma }-{\Gamma }_{\alpha \beta }^{\delta }\delta {\Gamma }_{\delta \gamma }^{\gamma }\hfill \\ & ={\left(\delta {\Gamma }_{\alpha }\right)}_{|\beta }-{\left(\delta {\Gamma }_{\alpha \beta }^{\gamma }\right)}_{|\gamma }.\hfill \end{array}$$

(76)

With respect to the Proposition 4 and the Equation (76), we get

$$\delta I_1=\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }\right)+\int d^N\sqrt{\left|g\right|}\left({\left(g^{\alpha \beta }\delta {\Gamma }_{\alpha }\right)}_{|\beta }-{\left(g^{\alpha \beta }\delta {\Gamma }_{\alpha \beta }^{\gamma }\right)}_{|\gamma }\right),$$

i.e.,

$$\delta I_1=\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }\right).$$

(77)

From the results presented in Proposition 4, we also conclude that

$$\begin{array}{ccc}\delta I_2=0,& \delta I_3=0,& \delta I_4=0.\end{array}$$

(78)

Based on Theorem 2, we recognize $g^{\pi \sigma }\to g^{\alpha \beta }$, $X_{\beta \gamma }^{\alpha }\to B_{\alpha \delta }^{\gamma }$, $Y_{ϵ\zeta }^{\delta }\to B_{\beta \gamma }^{\delta }$. With respect to the Equation (64), we make the forthcoming changes $U_{\beta \gamma \mu \nu }^{\alpha }\to {\tilde{\mathcal{B}}}_{\alpha \delta \mu \nu }^{\gamma }$, ${\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \theta }\to {ϵ}_{\alpha \delta \mu \nu }^{\gamma \theta }$, $V_{ϵ\zeta \mu \nu }^{\delta }\to {\tilde{\mathcal{B}}}_{\alpha \delta \mu \nu }^{\gamma }$, and ${\tilde{V}}_{ϵ\zeta \mu \nu }^{\alpha \delta }\to {\epsilon }_{\beta \delta \mu \nu }^{\gamma \theta }$. Hence, and with respect to the required symmetry by covariant indices $\mu$ and $\nu$, the next equality is satisfied

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }\right)& =\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }\left(B_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }+2B_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \zeta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \zeta })\right)g_{\mu \nu }\right)\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta }\hfill \\ & +{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta }.\hfill \end{array}$$

Based on this variation, and the Stokes Theorem, the variation of integral $I_5$ is

$$\begin{array}{cc}\hfill \delta I_5& =\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta })\hfill \\ & -\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta }.\hfill \end{array}$$

(79)

With respect to the Corollary 1 and the Equation (75), we recognize the next terms: $g^{\pi \sigma }\to g^{\alpha \beta }$, $X_{\beta \gamma }^{\alpha }\to B_{\alpha \beta }^{\gamma }$, and $Y_{\delta }\to {\Gamma }_{\gamma }$. Hence, we have $U_{\beta \gamma \mu \nu }^{\alpha }\to {\tilde{\mathcal{B}}}_{\alpha \beta \mu \nu }^{\gamma }$, ${\tilde{U}}_{\beta \gamma \mu \nu }^{\alpha \delta }\to {\epsilon }_{\alpha \delta \mu \nu }^{\gamma \theta }$, $V_{ϵ\zeta \mu \nu }^{\delta }\to -{\displaystyle \frac{1}{2}}{g}_{\mu \nu ,\gamma }$, ${\tilde{V}}_{\mu \nu }\to -{\displaystyle \frac{1}{2}}{g}_{\mu \nu }$. For this reason, the variation of $\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }$ is

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }\right)& =\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \nu }^{\alpha }{\Gamma }_{\alpha }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }{g}_{\mu \nu }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\gamma }{\Gamma }_{\delta }\left({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \delta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \delta }\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(({\tilde{\mathcal{B}}}_{\alpha \beta \mu \nu }^{\gamma }+{\tilde{\mathcal{B}}}_{\alpha \beta \nu \mu }^{\gamma }){\Gamma }_{\gamma }-B_{\alpha \beta }^{\gamma }{g}_{\mu \nu ,\gamma }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{\Gamma }_{\gamma }({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \theta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \theta })\right)}_{,\theta }+{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\gamma }{\epsilon }_{\alpha \beta \mu \nu }^{\gamma \theta }\right)}_{,\theta }.\hfill \end{array}$$

Hence, the variation of integral $I_6$ is

$$\begin{array}{cc}\hfill \delta I_6=& \int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \nu }^{\alpha }{\Gamma }_{\alpha }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \beta }^{\gamma }{\Gamma }_{\gamma }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\gamma }{\Gamma }_{\delta }\left({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \delta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \delta }\right)\hfill \\ & +{\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(({\tilde{\mathcal{B}}}_{\alpha \beta \mu \nu }^{\gamma }+{\tilde{\mathcal{B}}}_{\alpha \beta \nu \mu }^{\gamma }){\Gamma }_{\gamma }-B_{\alpha \beta }^{\gamma }{g}_{\mu \nu ,\gamma }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{\Gamma }_{\gamma }({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \theta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \theta })\right)}_{,\theta }.\hfill \end{array}$$

(80)

In the same manner as above, we get

$$\begin{array}{cc}\hfill \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }\right)& =\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\theta }{B}_{\alpha \delta }^{\gamma }\left({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta }\right)\hfill \\ & -\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta }\hfill \\ & +{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta },\hfill \\ \hfill \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }\right)& =\sqrt{\left|g\right|}\delta g^{\mu \nu }\left({\Gamma }_{\mu }{\Gamma }_{\nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }{g}_{\mu \nu }\right)-\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }{g}_{\mu \nu ,\beta }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }{g}_{\mu \nu }+\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }{g}_{\mu \nu }{\Gamma }_{\alpha }\right)}_{,\theta }-{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\alpha \theta }{\Gamma }_{\alpha }\right)}_{,\theta },\hfill \\ \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }\right)\hfill & \hfill ={\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }\left({\Gamma }_{\mu }{B}_{\nu \alpha }^{\alpha }+{\Gamma }_{\nu }{B}_{\mu \alpha }^{\alpha }-g^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }\right)-{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{g}_{\mu \nu ,\alpha }{B}_{\beta \delta }^{\delta }\\ & +{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }\left({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta }\right)+{\displaystyle \frac{1}{2}}\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }({\displaystyle \frac{1}{2}}{B}_{\alpha \gamma }^{\gamma }{g}_{\mu \nu }-{\Gamma }_{\alpha }{\tilde{\epsilon }}_{\mu \nu })\right)}_{,\theta }-\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }{\tilde{\epsilon }}_{\mu \nu }\hfill \\ & +{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \theta }({\Gamma }_{\alpha }{\tilde{\epsilon }}_{\mu \nu }-{\displaystyle \frac{1}{2}}{B}_{\alpha \gamma }^{\gamma }{g}_{\mu \nu })\right)}_{,\theta },\hfill \\ \delta \left(\sqrt{\left|g\right|}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }\right)\hfill & =\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \alpha }^{\alpha }{B}_{\nu \beta }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & +\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(B_{\alpha \gamma }^{\gamma }({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta })-2{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{\tilde{\epsilon }}_{\mu \nu }\right)\hfill \\ & -2\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }{B}_{\alpha \gamma }^{\gamma }{\tilde{\epsilon }}_{\mu \nu }\right)}_{,\theta }+2{\left(\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \theta }{B}_{\alpha \gamma }^{\gamma }{\tilde{\epsilon }}_{\mu \nu }\right)}_{,\theta }.\hfill \end{array}$$

With respect to Stokes’ Theorem, and the last four relations, one obtains that the variations of integrals $I_7$, $I_8$, $I_9$, and $I_{10}$ are

$$\begin{array}{cc}\hfill \delta I_7& =\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }{B}_{\beta \gamma }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta }-{\Gamma }_{\theta }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)\hfill \\ & -\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta })\right)}_{,\theta },\hfill \end{array}$$

(81)

$$\begin{array}{cc}\hfill \delta I_8& =\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left({\Gamma }_{\mu }{\Gamma }_{\nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }{g}_{\mu \nu }\right)\hfill \\ & -\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }{\Gamma }_{\alpha }\left({\Gamma }_{\mu \beta }^{\gamma }{g}_{\nu \gamma }+{\Gamma }_{\nu \beta }^{\gamma }{g}_{\mu \gamma }-{\Gamma }_{\beta }{g}_{\mu \nu }\right)\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }{g}_{\mu \nu }{\Gamma }_{\alpha }\right)}_{,\theta },\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill \delta I_9& ={\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left({\Gamma }_{\mu }{B}_{\nu \alpha }^{\alpha }+{\Gamma }_{\nu }{B}_{\mu \alpha }^{\alpha }-g^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(g_{\mu \nu ,\alpha }{B}_{\beta \delta }^{\delta }-{\Gamma }_{\alpha }({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta })-{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }+2{\Gamma }_{\alpha }{\Gamma }_{\beta }{\widetilde{ϵ}}_{\mu \nu }\right)\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }({\displaystyle \frac{1}{2}}{B}_{\alpha \gamma }^{\gamma }{g}_{\mu \nu }-{\Gamma }_{\alpha }{\tilde{\epsilon }}_{\mu \nu })\right)}_{,\theta },\hfill \end{array}$$

(83)

$$\begin{array}{cc}\hfill \delta I_{10}& =\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(B_{\mu \alpha }^{\alpha }{B}_{\nu \beta }^{\beta }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{B}_{\alpha \gamma }^{\gamma }{B}_{\beta \delta }^{\delta }{g}_{\mu \nu }\right)\hfill \\ & +\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(B_{\alpha \gamma }^{\gamma }({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta })-2{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{\tilde{\epsilon }}_{\mu \nu }\right)\hfill \\ & -2\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }{\left(g^{\alpha \theta }{B}_{\alpha \gamma }^{\gamma }{\tilde{\epsilon }}_{\mu \nu }\right)}_{,\theta }.\hfill \end{array}$$

(84)

After substituting the results (77)–(84) into the varied relation (70), one concludes that the variation of action (57) is

$$2\kappa \delta S_{mF}^{+}=\int d^{N}x\sqrt{\left|g\right|}\delta g^{\mu \nu }\left(R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }+{\rho }_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{\rho }_{\alpha \beta }{g}_{\mu \nu }+\tilde{\rho }{g}_{\mu \nu }+{\sigma }_{\mu \nu }+{\tau }_{\mu \nu ,\theta }^{\theta }\right),$$

(85)

where

$${\rho }_{\mu \nu }={\displaystyle \frac{N-1}{N+1}}\left(B_{\mu \beta }^{\alpha }{B}_{\nu \alpha }^{\beta }-B_{\mu \nu }^{\alpha }{\Gamma }_{\alpha }\right)-{\displaystyle \frac{N-1}{{(N+1)}^2}}\left({\Gamma }_{\mu }-B_{\mu \alpha }^{\alpha }\right)\left({\Gamma }_{\nu }-B_{\nu \beta }^{\beta }\right)$$

(86)

$$\tilde{\rho }=-{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{\Gamma }_{\beta }+{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta },$$

(87)

$$\begin{array}{cc}\hfill {\sigma }_{\mu \nu }& =g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta }\right)+{\displaystyle \frac{N-1}{2(N+1)}}{g}^{\alpha \beta }{\Gamma }_{\gamma }{\Gamma }_{\delta }\left({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \delta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \delta }\right)\hfill \\ & -{\displaystyle \frac{N-1}{2(N+1)}}{g}^{\alpha \beta }\left(({\tilde{\mathcal{B}}}_{\alpha \beta \mu \nu }^{\gamma }+{\tilde{\mathcal{B}}}_{\alpha \beta \nu \mu }^{\gamma }){\Gamma }_{\gamma }-B_{\alpha \beta }^{\gamma }({\Gamma }_{\mu \gamma }^{\delta }{g}_{\nu \delta }+{\Gamma }_{\nu \gamma }^{\delta }{g}_{\mu \delta })\right)\hfill \\ & -{\displaystyle \frac{2}{N+1}}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\tilde{\mathcal{B}}}_{\beta \gamma \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \gamma \nu \mu }^{\delta }-{\Gamma }_{\theta }({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta }\right)\hfill \\ & +{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }{\Gamma }_{\alpha }\left({\Gamma }_{\mu \beta }^{\gamma }{g}_{\nu \gamma }+{\Gamma }_{\nu \beta }^{\gamma }{g}_{\mu \gamma }-{\Gamma }_{\beta }{g}_{\mu \nu }\right)\hfill \\ & -{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }\left(({\Gamma }_{\mu \alpha }^{\gamma }{g}_{\nu \gamma }+{\Gamma }_{\nu \alpha }^{\gamma }{g}_{\mu \gamma })B_{\beta \delta }^{\delta }-{\Gamma }_{\alpha }({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta })+2{\Gamma }_{\alpha }{\Gamma }_{\beta }{\tilde{\epsilon }}_{\mu \nu }\right)\hfill \\ & -{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \beta }\left(B_{\alpha \gamma }^{\gamma }({\tilde{\mathcal{B}}}_{\beta \delta \mu \nu }^{\delta }+{\tilde{\mathcal{B}}}_{\beta \delta \nu \mu }^{\delta })-2{\Gamma }_{\alpha }{B}_{\beta \delta }^{\delta }{\tilde{\epsilon }}_{\mu \nu }\right),\hfill \end{array}$$

(88)

$$\begin{array}{cc}\hfill {\tau }_{\mu \nu }^{\theta }& =-g^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta }\right)+{\displaystyle \frac{N-1}{2(N+1)}}{g}^{\alpha \beta }{\Gamma }_{\gamma }\left({\epsilon }_{\alpha \beta \mu \nu }^{\gamma \theta }+{\epsilon }_{\alpha \beta \nu \mu }^{\gamma \theta }\right)\hfill \\ & +{\displaystyle \frac{2}{N+1}}{g}^{\alpha \beta }{B}_{\alpha \delta }^{\gamma }\left({\epsilon }_{\beta \gamma \mu \nu }^{\delta \theta }+{\epsilon }_{\beta \gamma \nu \mu }^{\delta \theta }\right)-{\displaystyle \frac{N-1}{{(N+1)}^2}}{g}^{\alpha \theta }{g}_{\mu \nu }{\Gamma }_{\alpha }\hfill \\ & +{\displaystyle \frac{2(N-1)}{{(N+1)}^2}}{g}^{\alpha \theta }\left({\displaystyle \frac{1}{2}}{B}_{\alpha \gamma }^{\gamma }{g}_{\mu \nu }-{\Gamma }_{\alpha }{\tilde{\epsilon }}_{\mu \nu }\right)+{\displaystyle \frac{2(N-1)}{{(N+1)}^2}}{g}^{\alpha \theta }{B}_{\alpha \gamma }^{\gamma }{\tilde{\epsilon }}_{\mu \nu }.\hfill \end{array}$$

(89)

From the equality (85), we get the Einstein’s equations

$$2\kappa T_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }+{\rho }_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{\rho }_{\alpha \beta }{g}_{\mu \nu }+\tilde{\rho }{g}_{\mu \nu }+{\sigma }_{\mu \nu }+{\tau }_{\mu \nu ,\theta }^{\theta },$$

(90)

for ${\rho }_{\mu \nu }$, $\tilde{\rho }$, ${\sigma }_{\mu \nu }$, and ${\tau }_{\mu \nu }^{\theta }$, given by (86), (87), (88), and (89), respectively.

With respect to a contravariant vector $u^{\alpha }$ and the invariant $W_{\mu \nu \sigma }^{\pi }$ given by (55), the compositions $E_{\nu }^{\pi }=W_{\mu \nu \sigma }^{\pi }{u}^{\mu }{u}^{\sigma }$ and $H_{\nu }^{\pi }={\displaystyle \frac{1}{2}}{{ϵ}_{\nu \alpha }}^{\beta \gamma }{W}_{\mu \beta \gamma }^{\pi }{u}^{\mu }{u}^{\alpha }$ are

$$\begin{array}{cc}\hfill E_{\nu }^{\pi }& =\left(R_{\alpha \nu \beta }^{\pi }+{\mathcal{Y}}_{\alpha \nu \beta }^{\pi }+{\mathcal{Z}}_{\alpha \nu \beta }^{\pi }-{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left((N+1)R_{\alpha \beta }+{\mathcal{X}}_{\alpha \beta }\right)\right)u^{\alpha }{u}^{\beta }\hfill \\ & +{\displaystyle \frac{1}{N^2-1}}\left((N+1)R_{\alpha \nu }+{\mathcal{X}}_{\alpha \nu }-(N-1){\mathcal{Y}}_{\nu \gamma \alpha }^{\gamma }\right)u^{\alpha }{u}^{\pi },\hfill \end{array}$$

(91)

$$H_{\nu }^{\pi }={\displaystyle \frac{1}{2}}{{ϵ}_{\nu \beta }}^{\gamma \delta }\left(R_{\alpha \gamma \delta }^{\pi }+{\mathcal{Y}}_{\alpha \gamma \delta }^{\pi }+{\mathcal{Z}}_{\alpha \gamma \delta }^{\pi }-{\displaystyle \frac{2}{N+1}}{\delta }_{\alpha }^{\pi }{\mathcal{Y}}_{\gamma ϵ|\delta }^{ϵ}\right)u^{\alpha }{u}^{\beta },$$

(92)

where ${\mathcal{X}}_{\mu \sigma }=N{B}_{\mu \alpha |\sigma }^{\alpha }+B_{\sigma \alpha |\mu }^{\alpha }-(N+1)B_{\mu \sigma |\alpha }^{\alpha }-(N+1)\left(B_{\mu \beta }^{\alpha }{B}_{\sigma \alpha }^{\beta }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \beta }^{\beta }\right)$, ${\mathcal{Y}}_{\mu \nu \sigma }^{\pi }=-B_{\mu \nu |\sigma }^{\pi }+B_{\mu \sigma |\nu }^{\pi }$, and ${\mathcal{Z}}_{\mu \nu \sigma }^{\pi }=B_{\mu \nu }^{\alpha }{B}_{\alpha \sigma }^{\pi }-B_{\mu \sigma }^{\alpha }{B}_{\alpha \nu }^{\pi }$.

6. Bridge Between Theory and Potential Applications

This research was conducted to define the new class of mappings of Riemannian spaces.

After that, the invariants for these mappings were obtained.

The basic invariant ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ given by (46) is non-vanishing. Using that, we defined an action which generalizes the corresponding Einstein–Hilbert one. The corresponding Einstein’s equations are obtained in this way.

Theoretically, we obtained another invariant $W_{\mu \nu \sigma }^{\pi }$ for the analyzed mappings, given by (55). This invariant is trace-free. Following this property, we created two compositions of this invariant analogous to the gravitoelectric and gravitomagnetic tensors.

Our research was realized for the special class of mappings of an N-dimensional Riemannian space. For $N=2$, our research may find applications in shell theory (2-dimensional manifolds defined with two spatial coordinates). For $N=3$, results of our research may be applied solid mechanics theory (3-dimensional manifolds defined with three spatial coordinates). Furthermore, it should be noted that the transformation from the initial to the current state of the curves directly reflects the evolution of particle acceleration within the considered manifold. This approach effectively replaces the classical notion of force with the geometric properties of the mapping, thereby establishing a fundamental link between the proposed theory and classical dynamics.

Example 1.

For $\overline{\rho }=\rho$, $\overline{m}=m=0$, and ${\overline{F}}_{\beta }^{\alpha }=F_{\beta }^{\alpha }=0$ in (24), the action (57) reduces to the well-known Einstein–Hilbert action $\int d^{4}x\sqrt{-g}{g}^{\alpha \beta }{R}_{\alpha \beta }$ The corresponding Einstein’s equations are $T_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{R}_{\alpha \beta }{g}_{\mu \nu }$.

Example 2.

For $\overline{\rho }=\rho$, $\overline{m}=m=0$, ${\overline{g}}^{\alpha \beta }{\overline{\sigma }}_{\alpha }{\overline{\sigma }}_{\beta }=g^{\alpha \beta }{\sigma }_{\alpha }{\sigma }_{\beta }=-1$, and ${\overline{F}}_{\alpha \beta }=-{\overline{F}}_{\beta \alpha }$ and $F_{\alpha \beta }=-F_{\beta \alpha }$, this model reduces to the Einstein–Maxwell one, $\int d^{4}x\sqrt{-g}(R-F^{\alpha \beta }{F}_{\alpha \beta })$. The corresponding Einstein’s equations are $R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}^{\alpha \beta }{R}_{\alpha \beta }{g}_{\mu \nu }=-{\displaystyle \frac{1}{{\mu }_0}}\left(F_{\mu \gamma }{F}_{\nu }^{\gamma }+{\displaystyle \frac{1}{4}}{F}_{\gamma \delta }{F}^{\gamma \delta }{g}_{\mu \nu }\right)$, for the magnet constant ${\mu }_0$.

The methodology employed in this study bridges the gap between the abstraction of theoretical research in the differential geometry of mappings and the concreteness of physical investigations. Specifically, the emergence of the divergence term ${\tau }_{\mu \nu ,\theta }^{\theta }$ in the derived field equations suggests that this geometric framework could provide a novel description of internal stresses in non-conservative systems or microstructural defects within the mechanics of solids and shells. Through this approach, actions are not extended artificially, but rather geometrically.

Following the methodology used in this research, other classes of mappings may be defined and studied in the same manner. For this reason, the methodology presented in this manuscript directly connects results in differential geometry of mappings and different aspects of research in physics.

7. Conclusions

In this theoretical consideration, research in differential geometry has been advanced to a clear level of potential applications in physics.

In Section 2, the enhanced method for obtaining invariants of mappings between Riemannian spaces, as presented in [12,13], is reviewed.

In Section 3, the concept of geodesic lines, generalized to m-conformal lines, is further extended to m-conformal F-lines. The $(\overline{m},m)$-conformal $(\overline{F},F)$-mappings of Riemannian spaces are defined. Furthermore, the relationship between the scalar functions $\overline{m}$ and m has been established, demonstrating that the first-order variation of the scalar function m is equal to 0.

In Section 4, the invariants of the mappings between Riemannian spaces defined in Section 3 are derived.

In Section 5, various geometric objects analogous to those utilized in physics are theoretically obtained.

In Section 6, it is clearly indicated where and how the results presented in Section 5 can be applied.