PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 98(112) (2015), 71–84 DOI: 10.2298/PIM1512071N

CONFORMAL AND GEODESIC MAPPINGS OF GENERALIZED EQUIDISTANT SPACES

Marija S. Najdanović, Milan Lj. Zlatanović, and Irena Hinterleitner

Abstract. We consider conformal and geodesic mappings of generalized equidistant spaces. We prove the existence of mentioned nontrivial mappings and construct examples of conformal and geodesic mapping of a 3-dimensional generalized equidistant space. Also, we find some invariant objects (three tensors and four which are not tensors) of special geodesic mapping of generalized equidistant space.

1. Introduction

Equidistant spaces are defined by the existence of concincular vector fields which are characterized by the property that their covariant derivative is proportional to the unity tensor. Examples of Riemannian spaces with concincular vector fields are the well known spatially homogeneous and isotropic cosmological models of space-time (pseudo-Riemannian manifolds with Friedmann–Lemaitre– Robertson–Walker metric) [7]. Equidistant spaces were studied in [1, 2, 6–11, 20, 21, 23, 24], etc.

The investigation of conformal and geodesic mapping theory for special spaces is an important and active research topic. Conformal mappings of Riemannian spaces with concincular vector fields were studied in the works of Brinkmann, Fialkov, Yano, de Vries. On the other hand, geodesic mappings of equidistant Riemannian spaces appeared in the papers of Sinjukov, Solodovnikov, Rosenfeld, Mikeš, Kiosak, Hall and many others.

In recent times, it has become very interesting to investigate spaces with nonsymmetric affine connection. The beginning of the study of general (nonsymmetric) affine connection spaces is especially related to the works [3,4] of Einstein on Unified Field Theory. Many new and interesting results related to generalized Riemannian spaces and, in general, nonsymmetric affine connection spaces, appeared in the papers of Eisenhart, Minčić, Nitescu, Prvanović, Stanković, Bohner, Yano, etc.

2010 Mathematics Subject Classification: 53B05.

Key words and phrases: generalized Riemannian space, equidistant space, conformal mapping, geodesic mapping, equitorsion geodesic mapping.

The authors gratefully acknowledge support from the research projects 174012 of the Serbian Ministry of Science and FAST-S-14-2346 of the Brno University of Technology.

Communicated by Stevan Pilipović.

For the first time, equidistant generalized Riemannian space was defined in [1] where geodesic mappings of such defined spaces were discussed. In the present paper we continue our previous investigations, primarily from [1] and [6], studying conformal and geodesic mappings of generalized equidistant spaces. Note that the study is of a local character. All functions considered are assumed to be sufficiently smooth.

1.1. Generalized Riemannian spaces. A generalized Riemannian space 𝔾ℝ_{ℕ} in the sense of Eisenhart’s definition [5] is a differentiable N-dimensional manifold, equipped with a nonsymmetric basic tensor g_{ij} (x), x = (x^{1}, …, x^{N}), where det(g_{ij} ) ≠ 0. We can write g_{ij} = g\underline{ij} + gij_{∨} where \underline{ij} denotes symmetriza-

tion and ij_{∨} antisymmetrization with division by indices i and j. The Riemannian

space ℝ_{ℕ} determined by the symmetric part of the metric tensor of generalized Riemannian space 𝔾ℝ_{ℕ}, is adjoint space of the space 𝔾ℝ_{ℕ}.

Generalized Christoffel symbols of the first kind of the space 𝔾ℝ_{ℕ} are given by

Γ_{i.jk} = \frac{1}{2}(g_{ji,k} − g_{jk,i} + g_{ik,j} ), i, j, k = 1, …, N,

where g_{ij,k} = \frac{∂g_{ij}}{∂x^{k}} . The connection coefficients of this space are generalized Christoffel symbols of the second kind Γ_{jk}^{i} = g\underline{ip}Γ_{p.jk} , where (g\underline{ij} ) = (g\underline{ij} )^{−1} , supposing det(g\underline{ij} ) ≠ 0. Generally, it is Γ_{jk}^{i} ≠ Γ_{kj}^{i} . Therefore, one can define the symmetric and anti-symmetric part of Γ_{jk}^{i}, respectively

Γi ; \underline{jk} = \frac{1}{2}(Γ_{jk}^{i} + Γ_{kj}^{i} ), Γi ; jk_{∨} = \frac{1}{2}(Γ_{jk}^{i} − Γ_{kj}^{i} ).

The magnitude Γi ; jk_{∨} is torsion tensor of the spaces 𝔾ℝ_{ℕ}. Obviously, Γ_{jk}^{i} = Γi ; \underline{jk} +Γi ; jk_{∨}.

Notice that in 𝔾ℝ_{ℕ} we have Γp ; ip_{∨} = 0 (eq. (2.10) in [15]).

Using the nonsymmetry of the connection Γ_{jk}^{i}, in the generalized Riemannian space, one can define four kinds of covariant derivatives (see [12–19]). For example, for a tensor a_{j}^{i}, we have

(1.1)

a_{j |_{1}m}^{i} = a_{j,m}^{i} + Γ_{pm}^{i}a_{j}^{p} − Γ_{jm}^{p} a_{p}^{i} , a_{j |_{2}m}^{i} = a_{j,m}^{i} + Γ_{mp}^{i}a_{j}^{p} − Γ_{mj}^{p} a_{p}^{i},

a_{j |_{3}m}^{i} = a_{j,m}^{i} + Γ_{pm}^{i}a_{j}^{p} − Γ_{mj}^{p} a_{p}^{i} , a_{j |_{4}m}^{i} = a_{j,m}^{i} + Γ_{mp}^{i}a_{j}^{p} − Γ_{jm}^{p} a_{p}^{i}.

Also, we can consider covariant derivative in 𝔾ℝ_{ℕ} with respect to the symmetric part of the connection Γ_{jk}^{i}. Thus,

(1.2)  a_{j;m}^{i} = a_{j,m}^{i} + Γi ; \underline{pm}a_{j}^{p} − Γp ; \underline{jm} a_{p}^{i}.

In the Riemannian space (Γi ; jk_{∨} = 0) all types of covariant derivates at (1.1) reduce

to (1.2).

In the case of the space 𝔾ℝ_{ℕ}, we have five independent curvature tensors [14] (in [14] R_{5} is denoted by R̃_{2}):

R_{1}_{jmn}^{i} = Γ_{jm,n}^{i} − Γ_{jn,m}^{i} + Γ_{jm}^{p} Γ_{pn}^{i} − Γ_{jn}^{p} Γ_{pm}^{i},
R_{2}_{jmn}^{i} = Γ_{mj,n}^{i} − Γ_{nj,m}^{i} + Γ_{mj}^{p} Γ_{np}^{i} − Γ_{nj}^{p} Γ_{mp}^{i},
R_{3}_{jmn}^{i} = Γ_{jm,n}^{i} − Γ_{nj,m}^{i} + Γ_{jm}^{p} Γ_{np}^{i} − Γ_{nj}^{p} Γ_{pm}^{i} + Γ_{nm}^{p} (Γ_{pj}^{i} − Γ_{jp}^{i}),
R_{4}_{jmn}^{i} = Γ_{jm,n}^{i} − Γ_{nj,m}^{i} + Γ_{jm}^{p} Γ_{np}^{i} − Γ_{nj}^{p} Γ_{pm}^{i} + Γ_{mn}^{p} (Γ_{pj}^{i} − Γ_{jp}^{i}),
R_{5}_{jmn}^{i} = \frac{1}{2}(Γ_{jm,n}^{i} + Γ_{mj,n}^{i} − Γ_{jn,m}^{i} − Γ_{nj,m}^{i} + Γ_{jm}^{p} Γ_{pn}^{i} + Γ_{mj}^{p} Γ_{np}^{i}
− Γ_{jn}^{p} Γ_{mp}^{i} − Γ_{nj}^{p} Γ_{pm}^{i})

These curvature tensors produce Ricci tensors of θ-kind, i.e., R_{θ}_{jmα}^{α} = R_{θ}_{jm} , θ ∈

{1, …, 5}.

2. Generalized equidistant spaces

Let 𝔾ℝ_{ℕ} be a generalized Riemannian space with a nonsymmetric metric tensor g_{ij} .

Definition 2.1. A vector field φ is called concircular if

(2.1)  φ_{;j}^{i} = ρδ_{j}^{i}.

where ρ is a function, δ_{j}^{i} is the Kronecker delta, (; ) denotes covariant derivative with respect to the symmetric part of the connection Γ_{jk}^{i}.

If ρ = const, φ is called convergent. A generalized Riemannian space 𝔾ℝ_{ℕ} with concircular vector field is called generalized equidistant space.

Condition (2.1) can be presented as φ_{i;j} = ρg\underline{ij} which means that the covariant derivative of φ_{i} (denoted by (;), (1.2)) is proportional to the symmetric part of the metric tensor of the space 𝔾ℝ_{ℕ}. Also, the previous condition can be presented as

φ_{i|_{1}j} = ρg\underline{ij} − Γp ; ij_{∨} φ_{p}, φ_{i|_{2}j} = ρg\underline{ij} − Γp ; ji_{∨} φ_{p}

where (|) denotes covariant derivative of the corresponding kind in the space 𝔾ℝ_{ℕ} and Γi ; pj_{∨} is the torsion tensor of 𝔾ℝ_{ℕ}.

It is known that in equidistant space ℝ_{ℕ} with symmetric metric tensor g_{ij} , where the concircular vector fields are nonisotropic (i.e., g_{ij} φ^{i}φ^{j} ≠ 0), we can introduce a system of the so-called canonical coordinates (x^{i}), where the metric is of the form

(2.2)  ds^{2} = a(x^{1})(dx^{1})^{2} + b(x^{1})ds̃^{2},

a, b ∈ C^{1} are nonzero functions, and ds̃^{2} = g̃_{σμ} (x^{2}, …, x^{N})dx^{σ} dx^{μ} is the metric form of certain Riemannian spaces ℝ̃_{ℕ}_{−1} (see [6]). Here, and in what follows, the indices σ, μ, θ … take values from 2 to N.

Let us look at the metric form of the space 𝔾ℝ_{ℕ}:

(2.3)  ds^{2} = g_{ij} dx^{i}dx^{j} = (g\underline{ij} + gij_{∨} )dx^{i}dx^{j} .

As it holds

g_{ij} dx^{i}dx^{j} = g_{ji} dx^{j} dx^{i} ⇔ (g_{ij} − g_{ji} )dx^{i}dx^{j} = 0 ⇔ gij_{∨} dx^{i}dx^{j} = 0,

we get that (2.3) becomes ds^{2} = g_{ij} dx^{i}dx^{j} = g\underline{ij} dx^{i}dx^{j}. So, we conclude that the basic metric form of 𝔾ℝ_{ℕ} can also be presented as (2.2). The symmetric parts of Christoffel symbols of the second kind satisfy:

Γ_{11}^{1} = \frac{1}{2} \frac{a′}{a}, Γ1 ; \underline{1σ} = Γ_{11}^{σ} = 0, Γ1 ; \underline{μσ} = −\frac{1}{2} \frac{b′}{a}g̃_{σμ} ,

Γμ ; \underline{σ1} = \frac{1}{2} \frac{b′}{b}δ_{σ}^{μ}, Γν ; \underline{σμ} = Γ̃_{σμ}^{ν} , (σ, μ, ν > 1),

where g̃_{σμ} are arbitrary symmetric functions of x^{2}, …, x^{N}, det(g̃_{σμ} ) ≠ 0, Γ̃_{σμ}^{ν} is Christoffel symbols of the second kind derived from g̃_{σμ} (x^{ν}).

Consider two generalized equidistant spaces 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ}, where the space 𝔾ℝ_{ℕ} has a metric form (2.2), and the space 𝔾\overline{ℝ}_{ℕ} has an analogous metric

(2.4)  ds̄^{2} = A(x^{1})(dx^{1})^{2} + B(x^{1})ĝ_{σμ} dx^{σ}dx^{μ},

where A, B ∈ C^{1} are nonzero functions, and ĝ_{σμ} are arbitrary symmetric functions of x^{2}, …, x^{N}, det(ĝ_{σμ} ) ≠ 0.

Let f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} be a mapping of two equidistant spaces. Consider the map in a common coordinate system x, i.e. the point M ∈ 𝔾ℝ_{ℕ} and its image f(M) ∈ 𝔾\overline{ℝ}_{ℕ} have the same coordinates x = (x^{1}, x^{2}, …, x^{N}). The corresponding geometric objects in 𝔾ℝ_{ℕ} will be marked with a bar. Then the symmetric part of the deformation tensor P_{jk}^{i} = \overline{Γ}_{jk}^{i} − Γ_{jk}^{i} of that mapping has the form

(2.5)  \begin{matrix}P_{11}^{1} = \frac{1}{2} ( \frac{A′}{A} − \frac{a′}{a}) , P 1 ; \underline{1σ} = P_{11}^{σ} = 0, P σ ; \underline{1μ} = \frac{1}{2} ( \frac{B′}{B} − \frac{b′}{b}) δ_{μ}^{σ}, \\ P 1 ; \underline{σμ} = −\frac{1}{2} ( \frac{B′}{A}ĝ_{σμ} − \frac{b′}{a}g̃_{σμ} ) , P σ ; \underline{μθ} = Γ̂_{μθ}^{σ} − Γ̃_{μθ}^{σ} \end{matrix}

The antisymmetric part of the deformation tensor will be denoted by ξ_{jk}^{i}, i.e.,

ξ_{jk}^{i} = P i ; jk_{∨} = \overline{Γ}i ; jk_{∨} − Γi ; jk_{∨}.

The following lemma holds for an arbitrary mapping of an arbitrary generalized Riemannian space.

Lemma 2.1. Under a mapping f of generalized Riemannian space 𝔾ℝ_{ℕ} onto generalized Riemannian space 𝔾\overline{ℝ}_{ℕ}, in the common coordinate system x with respect to the mapping, antisymmetric tensor ξ_{jk}^{i} satisfies

(2.6)  ξ_{ik}^{p} \overline{g}\underline{pj} + ξ_{jk}^{p} \overline{g}\underline{ip} = −Γp ; ik_{∨} \overline{g}\underline{pj} − Γp ; jk_{∨} \overline{g}\underline{ip},

where \overline{g}\underline{ij} is the symmetric part of the metric tensor of 𝔾\overline{ℝ}_{ℕ} and Γp ; ij_{∨} is the torsion

tensor of 𝔾ℝ_{ℕ}.

Proof. It is known that in the generalized Riemannian spaces holds \overline{g}_{\underline{i}j ||_{1}k} ≡ 0,

where we denote by ||_{1} covariant derivative of the first kind in 𝔾\overline{ℝ}_{ℕ} (see [12]). Using

the definition of covariant derivative, we get

(2.7)

\frac{∂\overline{g}\underline{ij}}{∂x^{k}} − \overline{Γ}_{ik}^{p} \overline{g}\underline{pj} − \overline{Γ}_{jk}^{p} \overline{g}\underline{ip} ≡ 0 ⇔ \frac{∂\overline{g}\underline{ij}}{∂x^{k}} − (\overline{Γ}p ; \underline{ik} + \overline{Γ}p ; ik_{∨})\overline{g}\underline{pj} − (\overline{Γ}p ; \underline{jk} + \overline{Γ}p ; jk_{∨})\overline{g}\underline{ip} ≡ 0

⇔ \overline{Γ}p ; ik_{∨} \overline{g}\underline{pj} + \overline{Γ}p ; jk_{∨} \overline{g}\underline{ip} ≡ 0,

where we used \overline{g}_{\underline{ij} ;k} ≡ 0, (; ) is covariant derivative in the adjoint Riemannian

space ℝ_{ℕ}. As it is ξ_{jk}^{i} = \overline{Γ}i ; jk_{∨} − Γi ; jk_{∨}, we get (2.6) from the last equation in (2.7). □

3. Conformal mappings of generalized equidistant spaces

Let 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ} be two generalized Riemannian spaces.

Definition 3.1. [22] The mapping f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} is conformal if in the common coordinate system x with respect to the mapping, the metric tensors g_{ij} and \overline{g}_{ij} of this spaces satisfy \overline{g}_{ij} = e^{2ψ} g_{ij} , where ψ is a function on 𝔾ℝ_{ℕ}.

For the Christoffel symbols of the first kind of 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ} the following relation is valid

\overline{Γ}_{i.jk} = e^{2ψ} (Γ_{i.jk} + g_{ji} ψ_{,k} − g_{jk} ψ_{,i} + g_{ik} ψ_{,j} )

and for the Christoffel symbols of the second kind

(3.1)  \overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + g\underline{ip}(g_{jp}ψ_{,k} − g_{jk} ψ_{,p} + g_{pk}ψ_{,j} ).

Let us introduce the notation ψ_{k} = ψ_{,k} = ∂ψ/∂x^{k} and ψ^{i} = g\underline{ip}ψ_{p}. From (3.1) we obtain

\overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + g\underline{ip}(g\underline{jp}ψ_{k} − g\underline{jk} ψ_{p} + g\underline{pk}ψ_{j}) + g\underline{ip}(gjp_{∨}ψ_{k} − gjk_{∨} ψ_{p} + gpk_{∨}ψ_{j}),

i.e.,

\overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i} ψ_{k} + δ_{k}^{i} ψ_{j} − ψ^{i}g\underline{jk} + ξ_{jk}^{i} , (ξ_{jk}^{i} = \overline{Γ}i ; jk_{∨} − Γi ; jk_{∨})

where

ξ_{jk}^{i} = g\underline{ip}(gjp_{∨}ψ_{k} − gjk_{∨} ψ_{p} + gpk_{∨}ψ_{j}) = −ξ_{kj}^{i} .

So, under conformal mappings, the deformation tensor satisfies

(3.2) P i ; \underline{jk} = δ_{j}^{i} ψ_{k} + δ_{k}^{i} ψ_{j} − ψ^{i}, g\underline{jk} , P i ; jk_{∨} = ξ_{jk}^{i} = g\underline{ip} (gjp_{∨} ψ_{k} − gjk_{∨} ψ_{p} + gpk_{∨} ψ_{j}).

By comparing (2.5) with the first equation in (3.2) for all cases of concrete values of indices i, j, k, we get

ψ_{1} = \frac{1}{2} ( \frac{A′}{A} − \frac{a′}{a}) = \frac{1}{2} ( \frac{B′}{B} − \frac{b′}{b}) = \frac{a}{2b}( \frac{B′}{A} − \frac{b′}{a}) , ψ_{σ} = 0, σ = 2, …, N,

wherefrom we obtain A(x^{1}) = ρ(x^{1})a(x^{1}), B(x^{1}) = ρ(x^{1})b(x^{1}) and

(3.3)  ψ = \frac{1}{2} ln|ρ| + c,

where ρ is an arbitrary function of x^{1}, ρ′ ≠ 0, and c is a constant. And also, according to (3.2), right, and (3.3), we get

(3.4)  ξ_{1σ}^{1} = ξ_{μθ}^{σ} = 0, ξ_{σμ}^{1} = −\frac{ρ′}{2aρ}gσμ_{∨} , ξ_{1μ}^{σ} = \frac{1}{2} \frac{ρ′}{ρ}g\underline{σρ} gρμ_{∨} .

The basic metric form of the space 𝔾\overline{ℝ}_{ℕ} is

(3.5)  ds̄^{2} = ρ(x^{1})( a(x^{1})(dx^{1})^{2} + b(x^{1})g̃_{σμ} dx^{σ}dx^{μ}) .

Thus, the following theorem holds

Theorem 3.1. Generalized equidistant space 𝔾ℝ_{ℕ} with fundament metric form (2.2) admits conformal mapping f on the generalized equidistant space 𝔾\overline{ℝ}_{ℕ} with fundament metric form (3.5), which is nontrivial for ρ′ ≠ 0, determined by a nonconstant ψ given in (3.3), and by the anti-symmetric tensor ξ_{jk}^{i} given by (3.4).

Example 3.1. Let the generalized Riemannian space 𝔾ℝ_{3} be given by the nonsymmetric matrix

(g_{ij} ) = [ \begin{matrix}1 & (x^{1})^{2} + (x^{3})^{2} & (x^{2})^{2} \\ −(x^{1})^{2} − (x^{3})^{2} & e^{x^{1}} x^{3} & e^{x^{1}} x^{2} + 1 \\ −(x^{2})^{2} & e^{x^{1}} x^{2} − 1 & e^{x^{1}} (x^{2} + x^{3}) \end{matrix} ]

Suppose that x^{2}x^{3}+(x^{3})^{2}−(x^{2})^{2} ≠ 0. Consider the symmetric and anti-symmetric part of the basic matrix, respectively

(g\underline{ij} ) = [ \begin{matrix}1 & 0 & 0 \\ 0 & e^{x^{1}} x^{3} & e^{x^{1}} x^{2} \\ 0 & e^{x^{1}} x^{2} & e^{x^{1}} (x^{2} + x^{3}) \end{matrix} ]

(gij_{∨} ) = [ \begin{matrix}0 & (x^{1})^{2} + (x^{3})^{2} & (x^{2})^{2} \\ −(x^{1})^{2} − (x^{3})^{2} & 0 & 1 \\ −(x^{2})^{2} & −1 & 0 \end{matrix} ]

Obviously, this space is equidistant and has the metric form (2.2) for a(x^{1}) = 1 and b(x^{1}) = e^{x^{1}} . The inverse matrix (g\underline{ij} ) = (g\underline{ij} )^{−1} is in the form

(g\underline{ij} ) = [ \begin{matrix}1 & 0 & 0 \\ 0 & \frac{x^{2} + x^{3}}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} & \frac{−x^{2}}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} \\ 0 & \frac{−x^{2}}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} & \frac{x^{3}}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} \end{matrix} ]

Let us construct a conformal mapping of the space 𝔾ℝ_{3}. According to the previous theorem, we can take ρ(x^{1}) = e^{x^{1}} , ρ′ ≠ 0, wherefrom ψ = \frac{1}{2}x^{1} for c = 0. From (3.4) we obtain

(3.6)  \begin{aligned}ξ_{1σ}^{1} & = ξ_{μθ}^{σ} = 0, ξ_{23}^{1} = −\frac{1}{2}, ξ_{13}^{2} = \frac{x^{2} + x^{3}}{2e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} , \\ ξ_{12}^{3} & = \frac{−x^{3}}{2e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} . \end{aligned}

Thus, the conformal mapping is determined by the deformation tensor

P_{jk}^{i} = δ_{j}^{i} ψ_{k} + δ_{k}^{i} ψ_{j} − ψ^{i}g\underline{jk} + ξ_{jk}^{i} ,

where ψ_{1} = \frac{1}{2}, ψ_{2} = ψ_{3} = 0, and ξ_{jk}^{i} is given by (3.6).

4. Geodesic mappings of generalized equidistant spaces

Let 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ} be two generalized Riemannian spaces.

Definition 4.1. [15] A diffeomorphism f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} is called geodesic mapping of 𝔾ℝ_{ℕ} onto 𝔾\overline{ℝ}_{ℕ} if f maps any geodesic curve in 𝔾ℝ_{ℕ} onto a geodesic curve in 𝔾\overline{ℝ}_{ℕ}.

According to [15,16], a necessary and sufficient condition that the mapping f is geodesic is that the deformation tensor has the form

(4.1)  P_{jk}^{i} = δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j} + ξ_{jk}^{i} ,

where

ψ_{i} = \frac{1}{1 + N} P p ; \underline{ip} = \frac{1}{1 + N} (\overline{Γ}p ; \underline{ip} − Γp ; \underline{ip}), (4.2)

ξ_{jk}^{i} = P i ; jk_{∨} = \overline{Γ}i ; jk_{∨} − Γi ; jk_{∨} . (4.3)

Equation (4.1) can be written as \overline{Γ}_{jk}^{i} = Γ_{jk}^{i} + δ_{j}^{i}ψ_{k} + δ_{k}^{i}ψ_{j} + ξ_{jk}^{i}.

Definition 4.2. [15] A geodesic mapping f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} is equitorsion if the torsion tensors of the spaces 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ} are equal in the corresponding points.

According to (4.3), it means that \overline{Γ}i ; jk_{∨} − Γi ; jk_{∨} = ξ_{jk}^{i} = 0.

Let us construct a geodesic mapping of generalized equidistant space 𝔾ℝ_{ℕ} with metric form (2.2) onto generalized equidistant space 𝔾\overline{ℝ}_{ℕ} with metric form (2.4). By comparing equations which describe symmetric part of deformation tensor (2.5) with necessary and sufficient condition (4.1) of the geodesic mapping we obtain

P_{11}^{1} = \frac{1}{2}( \frac{A′}{A} − \frac{a′}{a}) = ψ_{1}δ_{1}^{1} + ψ_{1}δ_{1}^{1} = 2ψ_{1}  ⇒ ψ_{1} = \frac{1}{4}( \frac{A′}{A} − \frac{a′}{a})
P 1 ; \underline{1σ} = 0 = ψ_{1}δ_{σ}^{1} + ψ_{σ}δ_{1}^{1} ⇒ ψ_{σ} = 0
P σ ; \underline{1μ} = \frac{1}{2}( \frac{B′}{B} − \frac{b′}{b}) δ_{μ}^{σ} = ψ_{1}δ_{μ}^{σ} + ψ_{μ}δ_{1}^{σ} = ψ_{1}δ_{μ}^{σ}  ⇒ ψ_{1} = \frac{1}{2}( \frac{B′}{B} − \frac{b′}{b})
P 1 ; \underline{σμ} = −\frac{1}{2}( \frac{B′}{A}ĝ_{σμ} − \frac{b′}{a}g̃_{σμ} ) = ψ_{σ}δ_{μ}^{1} + ψ_{μ}δ_{σ}^{1} = 0  ⇒ \frac{B′}{A} − \frac{b′}{a} = 0

From here, after some calculation, we obtain

A = \frac{pa(x^{1})}{(1 + qb(x^{1})^{2}} , B = \frac{pb(x^{1})}{1 + qb(x^{1})} ,

where p, q are constants such that p ≠ 0, 1 + qb(x^{1}) ≠ 0 and qb′(x^{1}) is not zero identically. The metric of 𝔾\overline{ℝ}_{ℕ} has the form

(4.4)  d\overline{s}^{2} = \frac{pa(x^{1})}{(1 + qb(x^{1})^{2}} (dx^{1})^{2} + \frac{pb(x^{1})}{1 + qb(x^{1})} g̃_{σμ} dx^{σ}dx^{μ}.

Also,

(4.5)  ψ = −\frac{1}{2} ln|1 + qb(x^{1})| + c,

where c is a constant. According to (2.6) and (4.4) for all cases of concrete values of indices i, j, k, we get the following system of equations

ξ_{σ1}^{1} = −Γ1 ; σ1_{∨} , (ξ_{1μ}^{ρ} + Γρ ; 1μ_{∨})\overline{g}\underline{ρσ} + (ξ_{σμ}^{1} + Γ1 ; σμ_{∨} )\overline{g}_{11} = 0,

(ξ_{σθ}^{ρ} + Γρ ; σθ_{∨})\overline{g}\underline{ρμ} + (ξ_{μθ}^{ρ} + Γρ ; μθ_{∨})\overline{g}\underline{σρ} = 0,

i.e.,

(4.6) \begin{matrix}ξ_{σ1}^{1} = −Γ1 ; σ1_{∨} , pb(x^{1})(1 + qb(x^{1}))(ξ_{1μ}^{ρ} + Γρ ; 1μ_{∨})g̃_{ρσ} + pa(x^{1})(ξ_{σμ}^{1} + Γ1 ; σμ_{∨} ) = 0, \\ (ξ_{σθ}^{ρ} + Γρ ; σθ_{∨})g̃_{ρμ} + (ξ_{μθ}^{ρ} + Γρ ; μθ_{∨})g̃_{σρ} = 0, \end{matrix}

wherefrom we can determine anti-symmetric tensor ξ_{jk}^{i}.

Thus, the following theorem holds

Theorem 4.1. Generalized equidistant space 𝔾ℝ_{ℕ} with fundament metric form (2.2) admits geodesic mapping f on the generalized equidistant space 𝔾\overline{ℝ}_{ℕ} with fundament metric form (4.4), which is nontrivial for p ≠ 0, 1+qb(x^{1}) ≠ 0 and nonzero qb′(x^{1}), determined by nonconstant ψ given in (4.5), and by the antisymmetric tensor ξ_{jk}^{i} given by (4.6).

Example 4.1. By the theorem given above, the function ψ = −\frac{1}{2} ln|1 + x^{1}| and the antisymmetric tensor

ξ_{21}^{1} = ξ_{31}^{1} = ξ_{23}^{2} = ξ_{23}^{3} = 0, ξ_{23}^{1} = x^{3} − x^{2},

ξ_{12}^{2} = −ξ_{13}^{3} = \frac{x^{2}(x^{2} − x^{3})}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} ,

ξ_{13}^{2} = \frac{(x^{2})^{2} − (x^{3})^{2}}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} , ξ_{12}^{3} = \frac{x^{3}(x^{3} − x^{2})}{e^{x^{1}} (x^{2}x^{3} + (x^{3})^{2} − (x^{2})^{2})} ,

determine a geodesic mapping of generalized equidistant space 𝔾ℝ_{3} given in the Example 3.1 onto equidistant space ℝ_{3}.

4.1. Invariant objects of equitorsion geodesic mapping. Invariant geometrical objects (invariants) are objects that do not change structure according to the corresponding mappings. In the case of geodesic mapping between two Riemannian spaces we have invariant geometric objects: the Tomas projective parameter and the Weyl projective tensor (see, for example, [10]). In [25] we found some new invariants according to the equitiorsion geodesic mappings f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ}. All these objects exist in the space 𝔾ℝ_{ℕ} and they are generalization of the Weyl projective tensor. Among five invariants, three of them are tensors and we called them “equitorsion projective tensors", and two of them are not tensors and we called them “equitorsion projective parameters". Using the condition of the equidistant spaces, we can find some interesting invariant geometrical objects which appear under equitorsion geodesic mapping of generalized equidistant spaces.

Let f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} be a geodesic mapping of the generalized Riemannian space which satisfies the condition

(4.7)  ψ_{ij} = ωg\underline{ij} ,

where ω is an invariant and ψ_{ij} = ψ_{i;j} −ψ_{i}ψ_{j}, or in the terms of covariant derivatives of the first and the second kinds

(4.8)  ψ_{1} _{ij} = ωg\underline{ij} − Γp ; ij_{∨} ψ_{p}, i.e. ψ_{2} _{ij} = ωg\underline{ij} + Γp ; ij_{∨} ψ_{p},

ψ_{k} _{ij} = ψ_{i|_{k}j} − ψ_{i}ψ_{j}, k = 1, 2. Then the space 𝔾ℝ_{ℕ} is generalized equidistant whose

equidistant congruence is generated by the vector ψ_{i}. Indeed, conditions (4.7) and (4.8) are equivalent to φ_{i;j} = ρg\underline{ij} in the case φ = e^{−ψ} , ρ = −ωe^{−ψ} and the last one presents the equation of generalized equidistant space. Further, suppose that the mapping f is equitorsion, i.e., the condition ξ_{jk}^{i} = 0 is in force. Let us find the invariant objects of this mapping of generalized equidistant space 𝔾ℝ_{ℕ}.

As it is known, the relations between the corresponding kinds of curvatures tensors of the spaces 𝔾ℝ_{ℕ} and 𝔾\overline{ℝ}_{ℕ} under equitorsion geodesic mapping are [22] (4.9)

\overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + δ_{j}^{i}(ψ_{1} _{mn} − ψ_{1} _{nm} ) + δ_{m}^{i} ψ_{1} _{jn} − δ_{n}^{i} ψ_{1} _{jm} + 2Γi ; mn_{∨} ψ_{j} + 2Γp ; mn_{∨} ψ_{p}δ_{j}^{i}
\overline{R}_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} + δ_{j}^{i}(ψ_{2} _{mn} − ψ_{2} _{nm} ) + δ_{m}^{i} ψ_{2} _{jn} − δ_{n}^{i} ψ_{2} _{jm} + 2Γi ; nm_{∨} ψ_{j} + 2Γp ; nm_{∨} ψ_{p}δ_{j}^{i}
\overline{R}_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + δ_{j}^{i}(ψ_{2} _{mn} − ψ_{1} _{nm} ) + δ_{m}^{i} ψ_{2} _{jn} − δ_{n}^{i} ψ_{1} _{jm} + 2Γi ; mj_{∨} ψ_{n} + 2Γi ; nj_{∨} ψ_{m}
\overline{R}_{4}_{jmn}^{i} = R_{4}_{jmn}^{i} + δ_{j}^{i}(ψ_{2} _{mn} − ψ_{1} _{nm} ) + δ_{m}^{i} ψ_{2} _{jn} − δ_{n}^{i} ψ_{1} _{jm} + 2Γi ; mj_{∨} ψ_{n} + 2Γi ; nj_{∨} ψ_{m}
\overline{R}_{5}_{jmn}^{i} = R_{5}_{jmn}^{i} + \frac{1}{2}δ_{j}^{i}(ψ_{1} _{mn} − ψ_{2} _{nm} + ψ_{2} _{mn} − ψ_{1} _{nm} ) + \frac{1}{2}δ_{m}^{i}(ψ_{1} _{jn} + ψ_{2} _{jn} )

− \frac{1}{2}δ_{n}^{i}(ψ_{1} _{jm} + ψ_{2} _{jm} ).

Let us start from the curvature tensor of the first kind. After using conditions (4.8) we obtain

(4.10) \overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + ω(δ_{m}^{i}g\underline{jn} − δ_{n}^{i}g\underline{jm} ) − (δ_{m}^{i}Γp ; jn_{∨} − δ_{n}^{i}Γp ; jm_{∨} )ψ_{p} + 2Γi ; mn_{∨} ψ_{j} .

Contracting by indices i and n in the previous equation we get

\overline{R}_{1}_{jm} = R_{1}_{jm} + ω(1 − N) − (1 − N)Γp ; jm_{∨} ψ_{p},

wherefrom we have

(4.11)  ωg\underline{jm} = \frac{1}{N − 1} (R_{1}_{jm} − \overline{R}_{1}_{jm} ) + Γp ; jm_{∨} ψ_{p}.

Put (4.11) into (4.10) and obtain

(4.12) \overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{1}_{jn} − \overline{R}_{1}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{1}_{jm} − \overline{R}_{1}_{jm} ) + 2Γi ; mn_{∨} ψ_{j} .

In the similar way, starting from the second equation in (4.9) and using (4.8) we obtain the following equation for the curvature tensor of the second kind

(4.13) \overline{R}_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{2}_{jn} − \overline{R}_{2}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{2}_{jm} − \overline{R}_{2}_{jm} ) + 2Γi ; nm_{∨} ψ_{j} .

Let us sum up (4.12) and (4.13). We get

\overline{R}_{1}_{jmn}^{i} + \overline{R}_{2}_{jmn}^{i} = R_{1}_{jmn}^{i} + R_{2}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} [R_{1}_{jn} + R_{2}_{jn} − (\overline{R}_{1}_{jn} + \overline{R}_{2}_{jn} )]

− \frac{δ_{n}^{i}}{N − 1} [R_{1}_{jm} + R_{2}_{jm} − (\overline{R}_{1}_{jm} + \overline{R}_{2}_{jm} )].

If we introduce the notation

Q_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + R_{2}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{1}_{jn} + R_{2}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{1}_{jm} + R_{2}_{jm} ),

we obtain \overline{Q}_{1}_{jmn}^{i} = Q_{1}_{jmn}^{i} , which means that Q_{1}_{jmn}^{i} is an invariant object of equitor-

sion geodesic mapping. Obviously, this object is a tensor.

Analogously to the previous consideration we can determine the relationships between the curvature tenors of the third, the fourth and the fifth kind under equitorsion geodesic mapping satisfying condition (4.8). Thus we have

(4.14)  \begin{matrix}\overline{R}_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{3}_{jn} − \overline{R}_{3}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{3}_{jm} − \overline{R}_{3}_{jm} ) \\ + 2δ_{m}^{i}Γp ; jn_{∨} ψ_{p} + 2ψ_{n}Γi ; mj_{∨} + 2ψ_{m}Γi ; nj_{∨} & , \\ \overline{R}_{4}_{jmn}^{i} = R_{4}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{4}_{jn} − \overline{R}_{4}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{4}_{jm} − \overline{R}_{4}_{jm} ) \\ + 2δ_{m}^{i}Γp ; jn_{∨} ψ_{p} + 2ψ_{n}Γi ; mj_{∨} + 2ψ_{m}Γi ; nj_{∨} & , \\ \overline{R}_{5}_{jmn}^{i} = R_{5}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{5}_{jn} − \overline{R}_{5}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{5}_{jm} − \overline{R}_{5}_{jm} ) & . \end{matrix}

After subtraction of the second equations from the first in (4.14) we obtain

\overline{R}_{3}_{jmn}^{i} − \overline{R}_{4}_{jmn}^{i} = R_{3}_{jmn}^{i} − R_{4}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} [R_{3}_{jn} − R_{4}_{jn} − (\overline{R}_{3}_{jn} − \overline{R}_{4}_{jn} )]

− \frac{δ_{n}^{i}}{N − 1} [R_{3}_{jm} − R_{4}_{jm} − (\overline{R}_{3}_{jm} − \overline{R}_{4}_{jm} )].

After introducing the denotation

Q_{2}_{jmn}^{i} = R_{3}_{jmn}^{i} − R_{4}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{3}_{jn} − R_{4}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{3}_{jm} − R_{4}_{jm} ),

we obtain \overline{Q}_{2}_{jmn}^{i} = Q_{2}_{jmn}^{i} , which presents the second invariant tensor of the map-

ping. From the last equation in (4.14) we get

Q_{3}_{jmn}^{i} = R_{5}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} R_{5}_{jn} − \frac{δ_{n}^{i}}{N − 1} R_{5}_{jm} ,

which is the third invariant tensor, i.e., \overline{Q}_{3}_{jmn}^{i} = Q_{3}_{jmn}^{i} .

Further, let us use (4.2) and the fact that the torsions of the corresponding spaces are equal under equitorsion mapping in (4.12). Then we have

\overline{R}_{1}_{jmn}^{i} = R_{1}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{1}_{jn} − \overline{R}_{1}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{1}_{jm} − \overline{R}_{1}_{jm} )

+ 2Γi ; mn_{∨} \frac{1}{N + 1} (\overline{Γ}p ; \underline{jp} − Γp ; \underline{jp})

= R_{1}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{1}_{jn} − \overline{R}_{1}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{1}_{jm} − \overline{R}_{1}_{jm} )

− \frac{2}{N + 1} Γi ; mn_{∨} Γp ; \underline{jp}.

Let us denote

Q_{4}_{jmn}^{i} = R_{1}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} R_{1}_{jn} − \frac{δ_{n}^{i}}{N − 1} R_{1}_{jm} − \frac{2}{N + 1} Γi ; mn_{∨} Γp ; \underline{jp}.

The object Q_{4}_{jmn}^{i} is invariant of the mapping, but is not a tensor. In the similar way we can write the second, the third and the fourth curvature tensor

\overline{R}_{2}_{jmn}^{i} = R_{2}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{2}_{jn} − \overline{R}_{2}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{2}_{jm} − \overline{R}_{2}_{jm} )

+ \frac{2}{N + 1} \overline{Γ}i ; nm_{∨} \overline{Γ}p ; \underline{jp} − \frac{2}{N + 1} Γi ; nm_{∨} Γp ; \underline{jp}.

\overline{R}_{3}_{jmn}^{i} = R_{3}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{3}_{jn} − \overline{R}_{3}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{3}_{jm} − \overline{R}_{3}_{jm} )

+ \frac{2}{N + 1} δ_{m}^{i}(\overline{Γ}p ; jn_{∨} \overline{Γ}q ; \underline{pq} − Γp ; jn_{∨} Γq ; \underline{pq})

+ \frac{2}{N + 1} (\overline{Γ}i ; mj_{∨} \overline{Γ}p ; \underline{np} − Γi ; mj_{∨} Γp ; \underline{np}) + \frac{2}{N + 1} (\overline{Γ}i ; nj_{∨} \overline{Γ}p ; \underline{mp} − Γi ; nj_{∨} Γp ; \underline{mp}),

\overline{R}_{4}_{jmn}^{i} = R_{4}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} (R_{4}_{jn} − \overline{R}_{4}_{jn} ) − \frac{δ_{n}^{i}}{N − 1} (R_{4}_{jm} − \overline{R}_{4}_{jm} )

+ \frac{2}{N + 1} δ_{m}^{i}(\overline{Γ}p ; jn_{∨} \overline{Γ}q ; \underline{pq} − Γp ; jn_{∨} Γq ; \underline{pq})

+ \frac{2}{N + 1} (\overline{Γ}i ; mj_{∨} \overline{Γ}p ; \underline{np} − Γi ; mj_{∨} Γp ; \underline{np}) + \frac{2}{N + 1} (\overline{Γ}i ; nj_{∨} \overline{Γ}p ; \underline{mp} − Γi ; nj_{∨} Γp ; \underline{mp})

wherefrom we get three more invariant parameters of the mapping which are not tensors:

Q_{5}_{jmn}^{i} = R_{2}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} R_{2}_{jn} − \frac{δ_{n}^{i}}{N − 1} R_{2}_{jm} − \frac{2}{N + 1} Γi ; nm_{∨} Γp ; \underline{jp},
Q_{6}_{jmn}^{i} = R_{3}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} R_{3}_{jn} − \frac{δ_{n}^{i}}{N − 1} R_{3}_{jm} − \frac{2}{N + 1} δ_{m}^{i}Γp ; jn_{∨} Γq ; \underline{pq}

− \frac{2}{N + 1} Γi ; mj_{∨} Γp ; \underline{np} − \frac{2}{N + 1} Γi ; nj_{∨} Γp ; \underline{mp},

Q_{7}_{jmn}^{i} = R_{4}_{jmn}^{i} + \frac{δ_{m}^{i}}{N − 1} R_{4}_{jn} − \frac{δ_{n}^{i}}{N − 1} R_{4}_{jm} − \frac{2}{N + 1} δ_{m}^{i}Γp ; jn_{∨} Γq ; \underline{pq}

− \frac{2}{N + 1} Γi ; mj_{∨} Γp ; \underline{np} − \frac{2}{N + 1} Γi ; nj_{∨} Γp ; \underline{mp}.

Thus, we proved the following theorem.

Theorem 4.2. Let f : 𝔾ℝ_{ℕ} → 𝔾\overline{ℝ}_{ℕ} be an equitorsion geodesic mapping of generalized equidistant space 𝔾ℝ_{ℕ} in which the equidistant congruence is generated by the vector ψ_{i} such that φ = e^{−ψ} . Then Q_{1}_{jmn}^{i} , Q_{2}_{jmn}^{i} , Q_{3}_{jmn}^{i} are invariant tensors and Q_{4}_{jmn}^{i} , Q_{5}_{jmn}^{i} , Q_{6}_{jmn}^{i} , Q_{7}_{jmn}^{i} are invariant parameters (not tensors) of this mapping.

5. Conclusion

The notion of generalized equidistant spaces first appeared in our paper [1]. We here continued the idea of generalized equidistant spaces, studying conformal and geodesic mappings of such spaces. We proved the existence of mentioned nontrivial mappings and constructed examples of conformal and geodesic mapping of a generalized equidistant space 𝔾ℝ_{3}.

Also, we found three invariant tensors and four invariant objects which are not tensors under geodesic mapping. By linear combinations of the obtained objects one can form new interesting invariant objects, but the question is how many of them are linearly independent and what they are.

The equidistant spaces are defined as the spaces satisfying the condition φ_{;j}^{i} = ρδ_{j}^{i}, for some vector field φ and a function ρ.

Due to the fact that in the generalized Riemannian spaces there are four kinds of covariant derivatives with respect to the connection Γ_{jk}^{i}, we can also define generalized equidistant spaces of the first and second kinds respectively by the conditions

φ_{|_{1}j}^{i} = ρδ_{j}^{i}, φ_{|_{2}j}^{i} = ρδ_{j}^{i}.

(Note that the covariant derivatives of the third and fourth kinds are reduced to the second and the first kind, respectively, in the case of φ^{i}.)

All this opens new questions and gives quite interesting ideas for the further investigation.

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Faculty of Sciences and Mathematics (Received 20 10 2014) University of Niš Serbia Preschool Teacher Training College Kruševac Serbia

marijamath@yahoo.com

Faculty of Sciences and Mathematics University of Niš Serbia

zlatmilan@pmf.ni.ac.rs

Department of Mathematics Faculty of Civil Engineering Brno University of Technology Czech Republic

Hinterleitner.Irena@seznam.cz