Equations of Motion and Navier–Stokes Equations

Abstract

In this research, we present the analogies between variational calculations in cosmology and in classical mechanics. Our approach is based on the invariants for transformations of affine connections defined on N-dimensional manifolds (special cases are the 8-dimensional, 5-dimensional, and 4-dimensional manifolds used in cosmology and 2-dimensional manifolds used in classical mechanics). Any of these transformations represents a class of curves on initial manifolds, which transmits to an another class of curves on the current manifolds. The main results of this paper are general equations of motion, which are obtained from the invariants caused by the transformation rule of an initial affine connection to the current one and the corresponding Navier–Stokes equations, recognized in transformations of curves along which moves a fluid particle.

1. Introduction

Differential geometry is a mathematical discipline that describes magnitudes that may help and support processes and theories in various sciences. For example, the well-known Einstein–Hilbert action $\int d^{4}x\sqrt{-g}R$, from which one obtains the energy–momentum tensor [1] of the form $T_{\mu \nu }=R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }$, has many different applications, such as in classical mechanics [2].

The tools for describing the objects are relatively complex (tensors as linear functions combined with commutators or indexed notations), so engineers usually stay in Euclidean spaces, where the position of indices is not important but results obtained from models are close to the experimental ones.

The purpose of this manuscript is to describe a differential geometric methodology that could be very useful for theoretical research, as well as for some computations in physics from cosmology to classical mechanics.

We will present universal equations of motion, which, depending on the dimensions of different Riemannian spaces, describe different general physical laws applied in different subjects of physics.

1.1. Riemannian Spaces

In this part of the manuscript, we will review some definitions regarding tensor calculus and Riemannian spaces that are necessary for this research. They may be found in [3,4,5] and in many other articles and monographs.

An N-dimensional manifold ${\mathcal{M}}_N={\mathcal{M}}_N(u^0,\dots ,u^{N-1})$ equipped with a symmetric metric tensor $\widehat{g}$, whose components are $g_{\mu \nu }$, $\mu ,\nu =0,\dots ,N-1$, is an N-dimensional Riemannian space ${\mathbb{R}}_N$. We assume it is $det\left[g_{\mu \nu }\right]\ne 0$. Hence, the contravariant metric tensor is defined by components as $\left[g^{\mu \nu }\right]={\left[g_{\mu \nu }\right]}^{-1}$.

The affine connection coefficients of space ${\mathbb{R}}_N$ are Christoffel symbols

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

(1)

where the comma denotes partial derivation, $g_{\mu \nu ,\pi }=\partial g_{\mu \nu }/\partial u^{\pi }$, and the Einstein summation convention is applied to the mute index $\alpha$.

Because an indexed value ${\tau }_{j_1\dots j_q}^{i_1\dots i_p}$ of the type $(p,q)$ is a tensor if under a change of reference frame $O{u}^{\alpha }$ to $O^{\prime }{u}^{\prime \alpha }$ its transformed value ${\tau }_{j_1^{\prime }\dots j_p^{\prime }}^{i_1^{\prime }\dots i_p^{\prime }}$ satisfies the equation

$${\tau }_{j_1^{\prime }\dots j_p^{\prime }}^{i_1^{\prime }\dots i_p^{\prime }}=x_{i_1}^{i_1^{\prime }}\dots x_{i_p}^{i_p^{\prime }}{x}_{j_1^{\prime }}^{j_1}\dots x_{j_q^{\prime }}^{j_q}{\tau }_{j_1\dots j_q}^{i_1\dots i_p},$$

(2)

where $x_{{\mu }^{\prime }}^{\pi }={\displaystyle \frac{\partial u^{\pi }}{\partial u^{{\mu }^{\prime }}}}$ and $x_{\mu }^{{\pi }^{\prime }}={\displaystyle \frac{\partial u^{{\pi }^{\prime }}}{\partial u^{\mu }}}$.

The Christoffell symbols ${\Gamma }_{\mu \nu }^{\pi }$ are not components of a tensor (of the type $(1,2)$) because the following relation holds:

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

(3)

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

Anyhow, the trace ${\Gamma }_{\mu }={\Gamma }_{\mu \alpha }^{\alpha }$ is a tensor of the type $(0,1)$ because

$${\Gamma }_{{\mu }^{\prime }}=x_{{\mu }^{\prime }}^{\mu }{\Gamma }_{\mu }+x_{\alpha }^{{\alpha }^{\prime }}{x}_{{\alpha }^{\prime }{\mu }^{\prime }}^{\alpha }=x_{{\mu }^{\prime }}^{\mu }{\Gamma }_{\mu }+x_{\alpha }^{{\alpha }^{\prime }}{\displaystyle \frac{\partial {\delta }_{\alpha }^{\alpha }}{\partial u^{{\mu }^{\prime }}}}=x_{{\mu }^{\prime }}^{\mu }{\Gamma }_{\mu }.$$

(4)

The covariant derivative of a tensor $a_{\mu }^{\pi }$ of the type $(1,1)$, with respect to the affine connection ${\Gamma }_{\mu \nu }^{\pi }$, is

$$a_{\mu |\nu }^{\pi }=a_{\mu ,\nu }^{\pi }+{\Gamma }_{\alpha \nu }^{\pi }{a}_{\mu }^{\alpha }-{\Gamma }_{\mu \nu }^{\alpha }{a}_{\alpha }^{\pi },$$

(5)

and it is a tensor of the type $(1,2)$.

With respect to the difference $a_{\mu \left|\nu \right|\sigma }^{\pi }-a_{\mu \left|\sigma \right|\nu }^{\pi }$, and the corresponding Ricci identity, the curvature tensor of space ${\mathbb{R}}_N$ is defined as

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

(6)

The corresponding Ricci tensor and scalar curvature of space ${\mathbb{R}}_N$ are

$$\begin{array}{ccc}{R}_{\mu \sigma }=R_{\mu \alpha \sigma }^{\alpha }& \mathrm{and}& R=g^{\alpha \beta }{R}_{\alpha \beta }.\end{array}$$

(7)

If ${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+P_{\mu \nu }^{\pi }$, where $P_{\mu \nu }^{\pi }$ is a tensor of the type $(1,2)$ symmetric by indices $\mu$ and $\nu$ and ${\overline{\Gamma }}_{\mu \nu }^{\pi }$ are Christoffell symbols of a space ${\overline{\mathbb{R}}}_N$, the curvature tensors ${\overline{R}}_{\mu \nu \sigma }^{\pi }$ of the space ${\overline{\mathbb{R}}}_N$ and $R_{\mu \nu \sigma }^{\pi }$ of the space ${\mathbb{R}}_N$ satisfy the relation

$$R_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }+P_{\mu \nu \parallel \sigma }^{\pi }-P_{\mu \sigma \parallel \nu }^{\pi }+P_{\mu \nu }^{\alpha }{P}_{\alpha \sigma }^{\pi }-P_{\mu \sigma }^{\alpha }{P}_{\alpha \nu }^{\pi },$$

(8)

where “||” denotes covariant derivative with respect to the affine connection ${\overline{\Gamma }}_{\mu \nu }^{\pi }$.

1.2. Research Purposes

We aim to obtain a geometrical object, invariant under a general transformation of an affine connection, whose trace is a monic polynomial of the Ricci tensor of Riemannian space.

In the next step, we will generalize the Einstein–Hilbert action $\int d^{4}x\sqrt{-g}R$ to the general form $\int d^{N}x\sqrt{-g}\mathcal{W}$, where a scalar $\mathcal{W}$ is the invariant from the previous paragraph composed of the contravariant metric $g^{\mu \nu }$.

In differential geometry, deformations of affine connections are mostly generated by transformation of a class of curves in an initial affine connection space to a curve from an another initial class of curves in the deformed affine connection space. In this research we will define special curves for deformations. Starting from these curves and their transformations, we will obtain the corresponding Navier–Stokes equations for a fluid that moves along the border of initial and deformed three-dimensional manifolds.

In this way, we will obtain a special geometrical object that will, depending on the dimension of internal geometry, connect two physical paradigms (vanishing of variations in actions and dynamics of particles connected in different models and kinds of physics).

2. Review on Invariants for Geometric Mappings

In this section, we will review the process for obtaining invariants of geometrical mappings.

For the transformations ${\overline{\Gamma }}_{\mu \nu }^{\pi }\to {\Gamma }_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }+P_{\mu \nu }^{\pi }$, where $P_{\mu \nu }^{\pi }={\delta }_{\mu }^{\pi }{\psi }_{\nu }+{\delta }_{\nu }^{\pi }{\psi }_{\mu }-g_{\mu \nu }{g}^{\pi \alpha }{\psi }_{\alpha }$, H. Weyl gave the well-known process for determination of the invariant from the corresponding transformation of curvature tensors ${\overline{R}}_{\mu \nu \sigma }^{\pi }\to R_{\mu \nu \sigma }^{\pi }$ [6]. Together with H. Weyl, T. Y. Thomas [7] obtained an invariant from the transformation ${\overline{\Gamma }}_{\mu \nu }^{\pi }\to {\Gamma }_{\mu \nu }^{\pi }$, but they did not find any other invariant from the corresponding transformation ${\overline{\mathbb{R}}}_{\mu \nu \sigma }^{\pi }\to R_{\mu \nu \sigma }^{\pi }$, the same as by using Weyl’s methodology. Mikeš and his research group [3,4], like Sinyukov [5], continued with the application of Weyl’s methodology for obtaining invariants for geometric mappings.

N. O. Vesić [8] preferred Weyl’s and Thomas’s methodology. He obtained one (associated) invariant of Thomas type and two (associated) invariants of Weyl type for the studied mappings. This methodology for obtaining invariants has been applied in many articles [9,10,11,12,13,14,15,16,17]. This methodology will be presented below.

Before we review the results regarding the invariants for the mappings between Riemannian spaces, we need to define an $\omega$-curve of the space ${\mathbb{R}}_N$. Namely, a curve $\ell =\ell \left(t\right)$ is an $\omega$-curve of the space ${\mathbb{R}}_N$ if its tangential vector ${\lambda }^{\pi }={\displaystyle \frac{d{\ell }^{\pi }}{dt}}$ satisfies the relation

$${\displaystyle \frac{d^2{\ell }^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\ell }^{\pi }}{dt}}+{\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}},$$

(9)

where ${\omega }_{\mu \nu }^{\pi }$ is a symmetric tensor of the type $(1,2)$.

A mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ that transmits any $\overline{\omega }$-curve of space ${\overline{\mathbb{R}}}_N$ to an $\omega$-curve of the space ${\mathbb{R}}_N$ is the $(\overline{\omega },\omega )$-mapping.

Starting from the definitions of a $\overline{\omega }$-curve and $\omega$-curve, i.e.,

$$\left\{\begin{array}{c}{\displaystyle \frac{d^2{\ell }^{\pi }}{d{t}^2}}+{\overline{\Gamma }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}=\overline{\rho }{\displaystyle \frac{d{\ell }^{\pi }}{dt}}+{\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}},\hfill \\ {\displaystyle \frac{d^2{\ell }^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\ell }^{\pi }}{dt}}+{\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}},\hfill \end{array}\right.$$

(10)

with respect to $\rho ={\rho }_{\beta }{\displaystyle \frac{d{\ell }^{\beta }}{dt}}$ and $\overline{\rho }={\overline{\rho }}_{\beta }{\displaystyle \frac{d{\ell }^{\beta }}{dt}}$, one gets

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

(11)

Consider a $(\overline{\omega },\omega )$-mapping $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ between the Riemannian spaces ${\overline{\mathbb{R}}}_N$ and ${\mathbb{R}}_N$. The basic equation of this mapping is (11). After contracting this equation by $\pi$ and $\nu$, and denoting ${\omega }_{\mu }={\omega }_{\mu \alpha }^{\alpha }$ and ${\overline{\omega }}_{\mu }={\overline{\omega }}_{\mu \alpha }^{\alpha }$, we obtain

$${\Gamma }_{\mu }={\overline{\Gamma }}_{\mu }+(N+1){\psi }_{\mu }+{\omega }_{\mu }-{\overline{\omega }}_{\mu }.$$

(12)

From the previous equation, we have

$${\psi }_{\mu }={\displaystyle \frac{1}{N+1}}{\Gamma }_{\mu }-{\displaystyle \frac{1}{N+1}}{\omega }_{\mu }-{\displaystyle \frac{1}{N+1}}{\overline{\Gamma }}_{\mu }+{\displaystyle \frac{1}{N+1}}{\overline{\omega }}_{\mu }.$$

(13)

After substituting the expression (13) into the basic Equation (11), we obtain

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

(14)

The relation (14) is equivalent to the equality ${\mathcal{T}}_{\mu \nu }^{\pi }={\overline{\mathcal{T}}}_{\mu \nu }^{\pi }$, where

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

(15)

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

(16)

are the associated basic invariants of Thomas type for the mapping f.

From the equality

$${\mathcal{W}}_{\mu \nu \sigma }^{\pi }:={\mathcal{T}}_{\mu \nu ,\sigma }^{\pi }-{\mathcal{T}}_{\mu \sigma ,\nu }^{\pi }+{\mathcal{T}}_{\mu \nu }^{\alpha }{\mathcal{T}}_{\alpha \sigma }^{\pi }-{\mathcal{T}}_{\mu \sigma }^{\alpha }{\mathcal{T}}_{\alpha \nu }^{\pi }={\overline{\mathcal{T}}}_{\mu \nu ,\sigma }^{\pi }-{\overline{\mathcal{T}}}_{\mu \sigma ,\nu }^{\pi }+{\overline{\mathcal{T}}}_{\mu \nu }^{\alpha }{\overline{\mathcal{T}}}_{\alpha \sigma }^{\pi }-{\overline{\mathcal{T}}}_{\mu \sigma }^{\alpha }{\overline{\mathcal{T}}}_{\alpha \nu }^{\pi }=:{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi },$$

we obtain the following basic associated invariants of Weyl type for the mapping f:

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

(17)

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

(18)

The geometrical objects ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ are tensors of the type $(1,3)$.

The invariants ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ may be expressed as

$$\begin{array}{cc}\hfill & {\mathcal{W}}_{\mu \nu \sigma }^{\pi }=R_{\mu \nu \sigma }^{\pi }+A_{\mu \nu \sigma }^{\pi }+{\delta }_{\mu }^{\pi }\left(X_{\nu \sigma }-X_{\sigma \nu }\right)+{\delta }_{\nu }^{\pi }{X}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{X}_{\mu \nu },\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }={\overline{R}}_{\mu \nu \sigma }^{\pi }+{\overline{A}}_{\mu \nu \sigma }^{\pi }+{\delta }_{\mu }^{\pi }\left({\overline{X}}_{\nu \sigma }-{\overline{X}}_{\sigma \nu }\right)+{\delta }_{\nu }^{\pi }{\overline{X}}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{\overline{X}}_{\mu \nu },\hfill \end{array}$$

(20)

where $A_{\mu \nu \sigma }^{\pi }=-{\omega }_{\mu \nu |\sigma }^{\pi }+{\omega }_{\mu \sigma |\nu }^{\pi }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha \sigma }^{\pi }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha \nu }^{\pi }$, ${\overline{A}}_{\mu \nu \sigma }^{\pi }=-{\overline{\omega }}_{\mu \nu \parallel \sigma }^{\pi }+{\overline{\omega }}_{\mu \sigma \parallel \nu }^{\pi }+{\overline{\omega }}_{\mu \nu }^{\alpha }{\overline{\omega }}_{\alpha \sigma }^{\pi }-{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha \nu }^{\pi }$, and the corresponding $X_{\mu \nu }$ and ${\overline{X}}_{\mu \nu }$.

The difference $0={\mathcal{W}}_{\mu \nu \sigma }^{\pi }-{\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$, for the invariants ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ and ${\overline{\mathcal{W}}}_{\mu \nu \sigma }^{\pi }$ given by (19) and (20), has the form

$$\begin{array}{ll}0=\left(R_{\mu \nu \sigma }^{\pi }-{\overline{R}}_{\mu \nu \sigma }^{\pi }\right)& +\left(A_{\mu \nu \sigma }^{\pi }-{\overline{A}}_{\mu \nu \sigma }^{\pi }\right)+{\delta }_{\mu }^{\pi }\left(X_{\nu \sigma }-X_{\sigma \nu }-{\overline{X}}_{\nu \sigma }+{\overline{X}}_{\sigma \nu }\right)\\ & +{\delta }_{\nu }^{\pi }\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right)-{\delta }_{\sigma }^{\pi }\left(X_{\mu \nu }-{\overline{X}}_{\mu \nu }\right).\end{array}$$

(21)

After contracting the last equality by $\pi$ and $\mu$, like by $\pi$ and $\nu$, and using $R_{\alpha \nu \sigma }^{\alpha }=0$ and ${\overline{R}}_{\alpha \nu \sigma }^{\alpha }=0$ in Riemannian spaces, we obtain the following results:

$$\begin{array}{cc}\hfill & 0=A_{\alpha \nu \sigma }^{\alpha }-{\overline{A}}_{\alpha \nu \sigma }^{\alpha }+(N+1)\left(X_{\nu \sigma }-X_{\sigma \nu }-{\overline{X}}_{\nu \sigma }+{\overline{X}}_{\sigma \nu }\right),\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & 0=R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }+A_{\mu \alpha \sigma }^{\alpha }-{\overline{A}}_{\mu \alpha \sigma }^{\alpha }+\left(X_{\mu \sigma }-X_{\sigma \mu }-{\overline{X}}_{\mu \sigma }+{\overline{X}}_{\sigma \mu }\right)+(N-1)\left(X_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\right).\hfill \end{array}$$

(23)

We obtain directly $A_{\alpha \nu \sigma }^{\alpha }=-{\omega }_{\nu |\sigma }+{\omega }_{\sigma |\nu }$, ${\overline{A}}_{\alpha \nu \sigma }^{\alpha }=-{\overline{\omega }}_{\nu \parallel \sigma }+{\overline{\omega }}_{\sigma \parallel \nu }$, $A_{\mu \alpha \sigma }^{\alpha }=-{\omega }_{\mu |\sigma }+{\omega }_{\mu \sigma |\alpha }^{\alpha }+{\omega }_{\mu \beta }^{\alpha }{\omega }_{\sigma \alpha }^{\beta }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha }$, and ${\overline{A}}_{\mu \alpha \sigma }^{\alpha }=-{\overline{\omega }}_{\mu \parallel \sigma }+{\overline{\omega }}_{\mu \sigma \parallel \alpha }^{\alpha }+{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\sigma \alpha }^{\beta }-{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha }$. If we substitute these expressions into the equalities (22) and (23), we obtain

$$\begin{array}{cc}\hfill & X_{\nu \sigma }-X_{\sigma \nu }-{\overline{X}}_{\nu \sigma }+{\overline{X}}_{\sigma \nu }={\displaystyle \frac{1}{N+1}}\left({\omega }_{\nu |\sigma }-{\omega }_{\sigma |\nu }\right)-{\displaystyle \frac{1}{N+1}}\left({\overline{\omega }}_{\nu \parallel \sigma }-{\overline{\omega }}_{\sigma \parallel \nu }\right),\hfill \end{array}$$

(24)

$$\begin{array}{ll}{X}_{\mu \sigma }-{\overline{X}}_{\mu \sigma }\hfill & =-{\displaystyle \frac{1}{N-1}}\left(R_{\mu \sigma }-{\overline{R}}_{\mu \sigma }\right)+{\displaystyle \frac{1}{N-1}}\left({\omega }_{\mu |\sigma }-{\omega }_{\mu \sigma |\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\sigma \alpha }^{\beta }+{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha }\right)\hfill \\ & -{\displaystyle \frac{1}{N-1}}\left({\overline{\omega }}_{\mu \parallel \sigma }-{\overline{\omega }}_{\mu \sigma \parallel \alpha }^{\alpha }-{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\sigma \alpha }^{\beta }+{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha }\right)\hfill \\ & -{\displaystyle \frac{1}{N^2-1}}\left({\omega }_{\mu |\sigma }-{\omega }_{\sigma |\mu }\right)+{\displaystyle \frac{1}{N^2-1}}\left({\overline{\omega }}_{\mu \parallel \sigma }-{\overline{\omega }}_{\sigma \parallel \mu }\right).\hfill \end{array}$$

(25)

The relations (24) and (25) transform the equality (21) into

$$\begin{array}{ll}0& =R_{\mu \nu \sigma }^{\pi }-{\overline{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 }\\ & +{\overline{\omega }}_{\mu \nu \parallel \sigma }^{\pi }-{\overline{\omega }}_{\mu \sigma \parallel \nu }^{\pi }-{\overline{\omega }}_{\mu \nu }^{\alpha }{\overline{\omega }}_{\alpha \sigma }^{\pi }+{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\omega }_{\nu |\sigma }-{\omega }_{\sigma |\nu }\right)\\ & -{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\overline{\omega }}_{\nu \parallel \sigma }-{\overline{\omega }}_{\sigma \parallel \nu }\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)\\ & +{\displaystyle \frac{1}{N-1}}{\delta }_{\nu }^{\pi }\left({\omega }_{\mu |\sigma }-{\omega }_{\mu \sigma |\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\sigma \alpha }^{\beta }+{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha }\right)\\ & -{\displaystyle \frac{1}{N-1}}{\delta }_{\sigma }^{\pi }\left({\omega }_{\mu |\nu }-{\omega }_{\mu \nu |\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\nu \alpha }^{\beta }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha }\right)\\ & -{\displaystyle \frac{1}{N-1}}{\delta }_{\nu }^{\pi }\left({\overline{\omega }}_{\mu \parallel \sigma }-{\overline{\omega }}_{\mu \sigma \parallel \alpha }^{\alpha }-{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\sigma \alpha }^{\beta }+{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha }\right)\\ & +{\displaystyle \frac{1}{N-1}}{\delta }_{\sigma }^{\pi }\left({\overline{\omega }}_{\mu \parallel \nu }-{\overline{\omega }}_{\mu \nu \parallel \alpha }^{\alpha }-{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\nu \alpha }^{\beta }+{\overline{\omega }}_{\mu \nu }^{\alpha }{\overline{\omega }}_{\alpha }\right)-{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left({\omega }_{\mu |\sigma }-{\omega }_{\sigma |\mu }\right)\\ & +{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left({\omega }_{\mu |\nu }-{\omega }_{\nu |\mu }\right)-{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left({\overline{\omega }}_{\mu \parallel \sigma }-{\overline{\omega }}_{\sigma \parallel \mu }\right)+{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left({\overline{\omega }}_{\mu \parallel \nu }-{\overline{\omega }}_{\nu \parallel \mu }\right).\end{array}$$

(26)

The relation (26) is equivalent to the equality $W_{\mu \nu \sigma }^{\pi }={\overline{W}}_{\mu \nu \sigma }^{\pi }$ for

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

(27)

$$\begin{array}{ll}{\overline{W}}_{\mu \nu \sigma }^{\pi }\hfill & ={\overline{R}}_{\mu \nu \sigma }^{\pi }-{\displaystyle \frac{1}{N-1}}\left({\delta }_{\nu }^{\pi }{\overline{R}}_{\mu \sigma }-{\delta }_{\sigma }^{\pi }{\overline{R}}_{\mu \nu }\right)\hfill \\ & -{\overline{\omega }}_{\mu \nu \parallel \sigma }^{\pi }+{\overline{\omega }}_{\mu \sigma \parallel \nu }^{\pi }+{\overline{\omega }}_{\mu \nu }^{\alpha }{\overline{\omega }}_{\alpha \sigma }^{\pi }-{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\overline{\omega }}_{\nu \parallel \sigma }-{\overline{\omega }}_{\sigma |\nu }\right)\hfill \\ & +{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left((N+1)\left({\overline{\omega }}_{\mu \parallel \sigma }-{\overline{\omega }}_{\mu \sigma \parallel \alpha }^{\alpha }-{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\sigma \alpha }^{\beta }+{\overline{\omega }}_{\mu \sigma }^{\alpha }{\overline{\omega }}_{\alpha }\right)-{\overline{\omega }}_{\mu \parallel \sigma }+{\overline{\omega }}_{\sigma \parallel \mu }\right)\hfill \\ & \hfill -{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left((N+1)\left({\overline{\omega }}_{\mu \parallel \nu }-{\overline{\omega }}_{\mu \nu \parallel \alpha }^{\alpha }-{\overline{\omega }}_{\mu \beta }^{\alpha }{\overline{\omega }}_{\nu \alpha }^{\beta }+{\overline{\omega }}_{\mu \nu }^{\alpha }{\overline{\omega }}_{\alpha }\right)-{\overline{\omega }}_{\mu \parallel \nu }+{\overline{\omega }}_{\nu \parallel \mu }\right).\end{array}$$

(28)

With the above, we have practically proved the following theorem.

Theorem 1.

Let $f:{\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ be a $(\overline{\omega },\omega )$-mapping. The geometrical object ${\mathcal{T}}_{\mu \nu }^{\pi }$ given by (15) is an invariant for this mapping. The geometrical object ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$ given by (17) is an invariant for the mapping f. The geometrical object $W_{\mu \nu \sigma }^{\pi }$ is also invariant for the mapping f.

The invariant $W_{\mu \nu \sigma }^{\pi }$ is an associated derived invariant for the mapping f. The traces of this invariant are $W_{\alpha \nu \sigma }^{\alpha }=0$, $W_{\mu \alpha \sigma }^{\alpha }=0$, $W_{\mu \nu \alpha }^{\alpha }=-W_{\mu \alpha \nu }^{\alpha }=0$.

Invariants for Mappings of Spaces ${\mathbb{R}}_2$ and ${\mathbb{R}}_3$

The surfaces in 3D Euclidean spaces are vector functions of two parameters. For this reason, their internal geometries correspond to the internal geometries of Riemannian spaces ${\mathbb{R}}_2$. They are two-dimensional manifolds ${\mathcal{M}}_2={\mathcal{M}}_2(u^0,u^1)$. In relativistic physics, it is most common to take $u^0=t$ and $u^1$ to be spatial parameters. In classical mechanics, both parameters $u^0$ and $u^1$ are spatial, but the time t is an outside parameter.

In both of these cases, the basic invariants ${\mathcal{T}}_{\mu \nu }^{\pi }$ and ${\mathcal{W}}_{\mu \nu \sigma }^{\pi }$, and the derived invariant $W_{\mu \nu \sigma }^{\pi }$ for a $(\overline{\omega },\omega )$-mapping $f:{\overline{\mathbb{R}}}_2\to {\mathbb{R}}_2$, are

$$\begin{array}{cc}\hfill & {\mathcal{T}}_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\omega }_{\mu \nu }^{\pi }-{\displaystyle \frac{1}{3}}\left({\Gamma }_{\nu }{\delta }_{\mu }^{\pi }+{\Gamma }_{\mu }{\delta }_{\nu }^{\pi }\right)+{\displaystyle \frac{1}{3}}\left({\omega }_{\nu }{\delta }_{\mu }^{\pi }+{\omega }_{\mu }{\delta }_{\nu }^{\pi }\right),\hfill \end{array}$$

(29)

$$\begin{array}{ll}{\mathcal{W}}_{\mu \nu \sigma }^{\pi }\hfill & =R_{\mu \nu \sigma }^{\pi }-{\omega }_{\mu \nu |\sigma }^{\pi }+{\omega }_{\mu \sigma |\nu }^{\pi }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha \sigma }^{\pi }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{3}}{\delta }_{\mu }^{\pi }\left({\omega }_{\nu |\sigma }-{\omega }_{\sigma |\nu }\right)\hfill \\ & -{\displaystyle \frac{1}{9}}{\delta }_{\nu }^{\pi }\left(3\left({\Gamma }_{\mu |\sigma }-{\omega }_{\mu |\sigma }+{\omega }_{\mu \sigma }^{\alpha }({\Gamma }_{\alpha }-{\omega }_{\alpha })\right)+\left({\Gamma }_{\mu }-{\omega }_{\mu }\right)\left({\Gamma }_{\sigma }-{\omega }_{\sigma }\right)\right)\hfill \\ & +{\displaystyle \frac{1}{9}}{\delta }_{\sigma }^{\pi }\left(3\left({\Gamma }_{\mu |\nu }-{\omega }_{\mu |\nu }+{\omega }_{\mu \nu }^{\alpha }({\Gamma }_{\alpha }-{\omega }_{\alpha })\right)+\left({\Gamma }_{\mu }-{\omega }_{\mu }\right)\left({\Gamma }_{\nu }-{\omega }_{\nu }\right)\right),\hfill \end{array}$$

(30)

$$\begin{array}{ll}{W}_{\mu \nu \sigma }^{\pi }\hfill & =R_{\mu \nu \sigma }^{\pi }-{\delta }_{\nu }^{\pi }{R}_{\mu \sigma }+{\delta }_{\sigma }^{\pi }{R}_{\mu \nu }\hfill \\ & -{\omega }_{\mu \nu |\sigma }^{\pi }+{\omega }_{\mu \sigma |\nu }^{\pi }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha \sigma }^{\pi }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{3}}{\delta }_{\mu }^{\pi }\left({\omega }_{\nu |\sigma }-{\omega }_{\sigma |\nu }\right)\hfill \\ & +{\displaystyle \frac{1}{3}}{\delta }_{\nu }^{\pi }\left(3\left({\omega }_{\mu |\sigma }-{\omega }_{\mu \sigma |\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\sigma \alpha }^{\beta }+{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha }\right)-{\omega }_{\mu |\sigma }+{\omega }_{\sigma |\mu }\right)\hfill \\ & -{\displaystyle \frac{1}{3}}{\delta }_{\sigma }^{\pi }\left(3\left({\omega }_{\mu |\nu }-{\omega }_{\mu \nu |\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\nu \alpha }^{\beta }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha }\right)-{\omega }_{\mu |\nu }+{\omega }_{\nu |\mu }\right),\hfill \end{array}$$

(31)

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

In classical mechanics, it is common to use the direct notation of the form $\underline{\underline{F}}\underline{\sigma }$, where the number of lines indicates the number of indices in operators $\widehat{F}$ and $\widehat{\sigma }$(for details, see [18]). The covariance or contravariance of indices is not relevant. The curvature of the space and affine connection coefficients are not implemented in the corresponding physical laws. The indices in $\underline{\underline{F}}$ and $\underline{\sigma }$ can have values 1, 2, and 3. The physical laws describe the action that deforms different objects.

Considering the geometrical literature, we conclude that this kind of classical mechanics describes the physics of a three-dimensional Euclidean space that surrounds different objects. The geometrical literature, i.e., the fact that tensors are defined on the borders of the manifolds, says that physical actions are defined on the border of a three-dimensional Euclidean space. The bad side of this approach is tyhe classical mechanics of leafs [19], flowers [20], etc., which when expressed by direct notation does not explain the internal physics of these objects.

The N-dimensional invariant ${\mathcal{W}}_{\mu \sigma }={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }$ will bridge this gap between the approaches. Moreover, we will obtain general equations of motion in next section, which, depending on the dimension of a manifold, express the equations of motion of a common form in cosmology and classical mechanics.

Anyhow, we will now present three invariants for a mapping f: ${\overline{\mathbb{R}}}_2\to {\mathbb{R}}_2$ in the case of a space ${\mathbb{R}}_2$ with a constant metric tensor. The relations will stay the same if the Riemannian space is a Euclidean space. It will be useful to obtain the corresponding results for a 3-dimensional Riemannian space with a constant metric.

If the metric of a space ${\mathbb{R}}_N$, $N=2,3$, is constant, the corresponding Christoffell symbols, curvature tensors, and Ricci tensors will vanish. The covariant derivative with respect to the affine connection of this space reduces to the corresponding partial derivative. Hence, in this case, the invariants for a mapping f: ${\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ are

$$\begin{array}{cc}\hfill & {\mathcal{T}}_{\mu \nu }^{\pi }=-{\omega }_{\mu \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}\left({\omega }_{\nu }{\delta }_{\mu }^{\pi }+{\omega }_{\mu }{\delta }_{\nu }^{\pi }\right),\hfill \end{array}$$

(32)

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

(33)

$$\begin{array}{ll}{W}_{\mu \nu \sigma }^{\pi }\hfill & =-{\omega }_{\mu \nu ,\sigma }^{\pi }+{\omega }_{\mu \sigma ,\nu }^{\pi }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha \sigma }^{\pi }-{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha \nu }^{\pi }+{\displaystyle \frac{1}{N+1}}{\delta }_{\mu }^{\pi }\left({\omega }_{\nu ,\sigma }-{\omega }_{\sigma ,\nu }\right)\hfill \\ & +{\displaystyle \frac{1}{N^2-1}}{\delta }_{\nu }^{\pi }\left((N+1)\left({\omega }_{\mu ,\sigma }-{\omega }_{\mu \sigma ,\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\sigma \alpha }^{\beta }+{\omega }_{\mu \sigma }^{\alpha }{\omega }_{\alpha }\right)-{\omega }_{\mu ,\sigma }+{\omega }_{\sigma ,\mu }\right)\hfill \\ & -{\displaystyle \frac{1}{N^2-1}}{\delta }_{\sigma }^{\pi }\left((N+1)\left({\omega }_{\mu ,\nu }-{\omega }_{\mu \nu ,\alpha }^{\alpha }-{\omega }_{\mu \beta }^{\alpha }{\omega }_{\nu \alpha }^{\beta }+{\omega }_{\mu \nu }^{\alpha }{\omega }_{\alpha }\right)-{\omega }_{\mu ,\nu }+{\omega }_{\nu ,\mu }\right).\hfill \end{array}$$

(34)

3. Equations of the Motion

Consider the N-dimensional Riemannian space ${\overline{\mathbb{R}}}_N$ (equipped with the affine connection ${\overline{\Gamma }}_{\mu \nu }^{\pi }$) and the space ${\mathbb{R}}_N$ (equipped with the affine connection ${\Gamma }_{\mu \nu }^{\pi }$). Let the deformation tensor $P_{\mu \nu }^{\pi }={\Gamma }_{\mu \nu }^{\pi }-{\overline{\Gamma }}_{\mu \nu }^{\pi }$ be of the form

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

(35)

for tensors ${\omega }_{\mu \nu }^{\pi }$ and ${\overline{\omega }}_{\mu \nu }^{\pi }$ of the type $(1,2)$ and symmetric by $\mu$ and $\nu$.

Based on the deformation tensor $P_{\mu \nu }^{\pi }$ given by (35) and the previous considerations, we conclude that the transformation ${\overline{\mathbb{R}}}_N\to {\mathbb{R}}_N$ is a $(\overline{\omega },\omega )$-mapping.

The invariant for this mapping, which is necessary for this section, is ${\mathcal{W}}_{\mu \sigma }={\mathcal{W}}_{\mu \alpha \sigma }^{\alpha }$. This invariant, with respect to the expression (17), has the form

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

(36)

We are going to eliminate variation from the following action:

$$C_0{S}_0^{+}=\int d^N\sqrt{\left|g\right|}{g}^{\alpha \beta }{\mathcal{W}}_{\alpha \beta }.$$

(37)

After contracting the deformation rule (35) by $\pi$ and $\nu$, we obtain that ${\psi }_{\mu }$ is given by Equation (13). For this reason, the Christoffell symbols ${\Gamma }_{\mu \nu }^{\pi }$ and ${\overline{\Gamma }}_{\mu \nu }^{\pi }$ satisfy Equation (14). This relation between the Christoffell symbols may be rewritten as

$${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }+d_{\mu \nu }^{\pi }-{\overline{d}}_{\mu \nu }^{\pi },$$

(38)

for

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

(39)

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

(40)

In this case, the invariant ${\mathcal{W}}_{\mu \sigma }$ becomes

$${\mathcal{W}}_{\mu \sigma }=R_{\mu \sigma }-d_{\mu |\sigma }+d_{\mu \sigma |\alpha }^{\alpha }+d_{\mu \beta }^{\alpha }{d}_{\sigma \alpha }^{\beta }-d_{\mu \sigma }^{\alpha }{d}_{\alpha }.$$

(41)

Before we continue the further calculations, we need to recall that ${\Gamma }_{\mu }={\displaystyle \frac{{\sqrt{\left|g\right|}}_{,\mu }}{\sqrt{\left|g\right|}}}$, i.e., ${\sqrt{\left|g\right|}}_{,\mu }=\sqrt{\left|g\right|}{\Gamma }_{\mu }$. For this reason, for any tensor ${\tau }^{\alpha }$ of the type $(1,0)$, we obtain

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

(42)

Based on the assumption that $\delta g^{\mu \nu }$ is equal 0 at the border of integration, and with respect to the result (42) combined with Stokes’ theorem, we conclude

$$\delta \left(\int d^{N}x\sqrt{\left|g\right|}{\tau }_{|\alpha }^{\alpha }\right)=0.$$

(43)

Hence, the variation in action (37) for $g<0$ reduces to

$$C_0\delta S_0^{+}=\delta \left(\int d^{N}x\sqrt{-g}R\right)+\delta \left(\int d^{N}x\sqrt{-g}{g}^{\alpha \beta }\left(d_{\alpha \delta }^{\gamma }{d}_{\beta \gamma }^{\delta }-d_{\alpha \beta }^{\gamma }{d}_{\gamma }\right)\right).$$

(44)

By the quotient rule, there exists a tensor ${\mathcal{D}}_{\beta \gamma \mu \nu }^{\alpha }$ of the type $(1,4)$ such that the variation in $d_{\beta \gamma }^{\alpha }$ is expressed as $\delta d_{\beta \gamma }^{\alpha }={\mathcal{D}}_{\beta \gamma \mu \nu }^{\alpha }\delta g^{\mu \nu }$. Moreover, the variation in $\delta d_{\alpha }$ is $\delta d_{\alpha }={\mathcal{D}}_{\alpha \beta \mu \nu }^{\beta }\delta g^{\mu \nu }$. It is well known what the variation in the first summand is in the previous equation. Hence

$$\begin{array}{ll}{C}_0\delta S_0^{+}& =\int d^{N}x\sqrt{-g}\delta g^{\mu \nu }\left(R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }\right)+\int d^{N}x\sqrt{-g}\delta g^{\mu \nu }\left(d_{\mu \beta }^{\alpha }{d}_{\nu \alpha }^{\beta }-d_{\mu \nu }^{\alpha }{d}_{\alpha }\right)\\ & -{\displaystyle \frac{1}{2}}\int d^{N}x\sqrt{-g}\delta g^{\mu \nu }{g}_{\mu \nu }{g}^{\alpha \beta }\left(d_{\alpha \delta }^{\gamma }{d}_{\beta \gamma }^{\delta }-d_{\alpha \beta }^{\gamma }{d}_{\gamma }\right)\\ & +\int d^{N}x\sqrt{-g}\delta g^{\mu \nu }{g}^{\alpha \beta }\left(2{\mathcal{D}}_{\alpha \delta \mu \nu }^{\gamma }{d}_{\beta \gamma }^{\delta }-{\mathcal{D}}_{\alpha \beta \mu \nu }^{\gamma }{d}_{\gamma }-d_{\alpha \beta }^{\gamma }{\mathcal{D}}_{\gamma \delta \mu \nu }^{\delta }\right).\end{array}$$

(45)

After involving the Lagrangian density ${\mathcal{L}}_M$ in the action (37), i.e., if we analyze the action $\int d^{N}x\sqrt{-g}\left(g^{\alpha \beta }{\mathcal{W}}_{\alpha \beta }+{\mathcal{L}}_M\right)$, the energy–momentum tensor is

$$\begin{array}{ll}{T}_{\mu \nu }& =R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }+d_{\mu \beta }^{\alpha }{d}_{\nu \alpha }^{\beta }-d_{\mu \nu }^{\alpha }{d}_{\alpha }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }{g}^{\alpha \beta }\left(d_{\alpha \delta }^{\gamma }{d}_{\beta \gamma }^{\delta }-d_{\alpha \beta }^{\gamma }{d}_{\gamma }\right)\\ & +g^{\alpha \beta }\left(2{\mathcal{D}}_{\alpha \delta \mu \nu }^{\gamma }{d}_{\beta \gamma }^{\delta }-{\mathcal{D}}_{\alpha \beta \mu \nu }^{\gamma }{d}_{\gamma }-d_{\alpha \beta }^{\gamma }{\mathcal{D}}_{\gamma \delta \mu \nu }^{\delta }\right).\end{array}$$

(46)

The equation of motion (46) holds for any dimension N, $2\le N\le +\infty$. The summand in the last row in (46) is a generalization of Einstein’s cosmological constant $\Lambda$.

4. Generalized Navier–Stokes Equations

In the previous section, we obtained physical laws by vanishing the variation in the generalized action. The action is generated by the basic Equation (14).

In the case of dimension $N=2$, the basic Equation (14) means that any $\overline{\omega }$-curve of the space ${\overline{\mathbb{R}}}_2$ is transformed to an $\omega$-curve of the space ${\mathbb{R}}_2$.

In our example, let a fluid particle moves along the $\overline{\omega }$-curve of the space ${\overline{\mathbb{R}}}_2$. After the deformation of the surface, the $\overline{\omega }$-curve is transformed to an $\omega$-curve of the space ${\mathbb{R}}_2$.

The equations of the $\overline{\omega }$-curve of space ${\overline{\mathbb{R}}}_2$ and $\omega$-curve of space ${\mathbb{R}}_2$ are

$$\begin{array}{c}\hfill {\displaystyle \frac{d^2{\overline{\ell }}^{\pi }}{d{t}^2}}+{\overline{\Gamma }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\overline{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\overline{\ell }}^{\beta }}{dt}}=\overline{\rho }{\displaystyle \frac{d{\overline{\ell }}^{\pi }}{dt}}+{\overline{\omega }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\overline{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\overline{\ell }}^{\beta }}{dt}},\end{array}$$

(47)

$$\begin{array}{c}\hfill {\displaystyle \frac{d^2{\ell }^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}=\rho {\displaystyle \frac{d{\ell }^{\pi }}{dt}}+{\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}.\end{array}$$

(48)

The corresponding Navier–Stokes equations are

$$\left\{\begin{array}{c}{\displaystyle \frac{d^2{\overline{\ell }}^{\pi }}{d{t}^2}}+{\overline{\Gamma }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\overline{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\overline{\ell }}^{\beta }}{dt}}=-{\overline{g}}^{\pi \alpha }{\overline{p}}_{\parallel \alpha }+{\overline{f}}_{diss}^{\pi }+{\overline{\omega }}_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\overline{\ell }}^{\alpha }}{dt}}{\displaystyle \frac{d{\overline{\ell }}^{\beta }}{dt}},\hfill \\ {\displaystyle \frac{d^2{\ell }^{\pi }}{d{t}^2}}+{\Gamma }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}=-g^{\pi \alpha }{p}_{|\alpha }+f_{diss}^{\pi }+{\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}.\hfill \end{array}\right.$$

(49)

The left sides of the last two Navier–Stokes equations are covariant accelerations of the particle. The values $-{\overline{g}}^{\pi \alpha }{\overline{p}}_{\parallel \alpha }$ and $-g^{\pi \alpha }{p}_{|\alpha }$ are gradients of pressures $\overline{p}$ and p. The tensors ${\overline{f}}_{diss}^{\pi }$ and $f_{diss}^{\pi }$ are components of viscosity (if they exist). The symmetric tensors ${\overline{\omega }}_{\alpha \beta }^{\pi }$ and ${\omega }_{\alpha \beta }^{\pi }$ may be factored as ${\overline{\omega }}_{\alpha \beta }^{\pi }={\overline{g}}^{\pi \gamma }\left({\overline{F}}_{\gamma \alpha }{\overline{\sigma }}_{\beta }+{\overline{F}}_{\gamma \beta }{\overline{\sigma }}_{\alpha }+{\overline{\theta }}_{\alpha \beta }{\overline{\phi }}_{\gamma }\right)$ and ${\omega }_{\alpha \beta }^{\pi }=g^{\pi \gamma }\left(F_{\gamma \alpha }{\sigma }_{\beta }+F_{\gamma \beta }{\sigma }_{\alpha }+{\theta }_{\alpha \beta }{\phi }_{\gamma }\right)$, where ${\overline{F}}_{\alpha \beta }$ and $F_{\alpha \beta }$ are non-symmetric tensors of the type $(0,2)$, ${\overline{\theta }}_{\alpha \beta }$ and ${\theta }_{\alpha \beta }$ are symmetric tensors of the type $(0,2)$, and ${\sigma }_{\alpha }$, ${\overline{\sigma }}_{\alpha }$, ${\overline{\phi }}_{\alpha }$, and ${\phi }_{\alpha }$ are covariant vectors. These vectors may be expressed as ${\overline{\sigma }}_{\alpha }={\overline{\sigma }}_{\parallel \alpha }$, ${\sigma }_{\alpha }={\sigma }_{|\alpha }$, ${\overline{\phi }}_{\alpha }={\overline{\phi }}_{\parallel \alpha }$, and ${\phi }_{\alpha }={\phi }_{|\alpha }$ for fields $\overline{\sigma }$, $\sigma$, $\overline{\phi }$, and $\phi$.

Let $d^2{\overline{\ell }}^{\pi }/d{t}^2={\overline{a}}^{\pi }$, $d{\ell }^{\pi }/d{t}^2=a^{\pi }$, $d{\overline{\ell }}^{\pi }/dt={\overline{v}}^{\pi }$, and $d{\ell }^{\pi }/dt=v^{\pi }$. The Navier–Stokes Equation (49) transforms to

$$\left\{\begin{array}{c}{\overline{a}}^{\pi }+{\overline{\Gamma }}_{\alpha \beta }^{\pi }{\overline{v}}^{\alpha }{\overline{v}}^{\beta }=-{\overline{g}}^{\pi \alpha }{\overline{p}}_{\parallel \alpha }+{\overline{f}}_{diss}^{\pi }+{\overline{\omega }}_{\alpha \beta }^{\pi }{\overline{v}}^{\alpha }{\overline{v}}^{\beta },\hfill \\ a^{\pi }+{\Gamma }_{\alpha \beta }^{\pi }{v}^{\alpha }{v}^{\beta }=-g^{\pi \alpha }{p}_{|\alpha }+f_{diss}^{\pi }+{\omega }_{\alpha \beta }^{\pi }{v}^{\alpha }{v}^{\beta }.\hfill \end{array}\right.$$

(50)

The vectors ${\overline{a}}^{\pi }$, $a^{\pi }$, ${\overline{v}}^{\pi }$, and $v^{\pi }$ correspond to the components of the acceleration (first two vectors) and the velocity (last two vectors) in classical mechanics. The geometrical objects ${\overline{\Gamma }}_{\alpha \beta }^{\pi }{\overline{v}}^{\alpha }{\overline{v}}^{\beta }$ and ${\Gamma }_{\alpha \beta }^{\pi }{v}^{\alpha }{v}^{\beta }$ are kinematic terms that represent the geodetic acceleration. They describe the change in velocity due to the curvature of the manifold.

Let us recognize the Newtonian mechanics in Equation (50). Newtonian mechanics corresponds to the case of $g_{\alpha \beta }={\delta }_{\alpha \beta }$, and analogously for ${\overline{g}}_{\alpha \beta }={\delta }_{\alpha \beta }$. In this case, ${\Gamma }_{\mu \nu }^{\pi }={\overline{\Gamma }}_{\mu \nu }^{\pi }=0$. Newtonian mechanics, i.e., its most simple case, describes movement without actions of additional forces. For this reason, it holds that $-g^{\pi \alpha }{p}_{|\alpha }=0$ and $f_{diss}^{\pi }=0$. After this, we obtain $a^{\pi }={\omega }_{\alpha \beta }^{\pi }{\displaystyle \frac{d{\ell }^{\alpha }}{dt}}{\displaystyle \frac{d{\ell }^{\beta }}{dt}}$. The right side of the last equality, which is equal to the force $F^{\pi }$ by mass m, is equivalent to the second Newtonian law $m{a}^{\pi }=F^{\pi }$. In the same manner, we get $\overline{m}{\overline{a}}^{\pi }={\overline{F}}^{\pi }$ from the first equation in (50).

These expressions are valid for any dimension N, but we aimed to present the case of $N=2$.

5. Conclusions

In this research, we discovered a new common point of view between differential geometry and classical mechanics. To connect these two subjects, we needed to review Vesić’s methodology for obtaining invariants of geometric mappings (Section 2).

Using the new invariant ${\mathcal{W}}_{\mu \sigma }$, we generalized the Einstein–Hilbert action in an N-dimensional Riemannian space. This result makes it possible to obtain equations of motion in any dimensional space (Section 3).

As a special case, we generalized Navier–Stokes equations for a fluid in a $3D$ manifold. In our approach, we defined forces on this manifold, not at the border of Euclidean space. The Navier–Stokes equations are generalized with respect to the affine connection of the manifold, not Euclidean space (Section 4).

Regarding some possible new research in this field, as a future perspective, it would be possible to analyze the Navier–Stokes-type equations of motion, taking into account the presence of the torsion field. In other words, this is useful for an Einstein–Cartan geometry or its extensions.