Geodesic Mappings of V_n(K)-Spaces and Concircular Vector Fields

Abstract

In the present paper, we study geodesic mappings of special pseudo-Riemannian manifolds called $V_n\left(K\right)$-spaces. We prove that the set of solutions of the system of equations of geodesic mappings on $V_n\left(K\right)$-spaces forms a special Jordan algebra and the set of solutions generated by concircular fields is an ideal of this algebra. We show that pseudo-Riemannian manifolds admitting a concircular field of the basic type form the class of manifolds closed with respect to the geodesic mappings.

1. Introduction

The problem of geodesic mappings of the pseudo-Riemannian manifold was first studied by Levi-Civita [1]. There exist many monographs and papers devoted to the theory of geodesic mappings and transformations [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]. Geodesic mappings play an important role in the general theory of relativity [8,26].

Let $A_n=(M_n,\nabla )$ be an n-dimensional manifold $M_n$ with an affine connection ∇ without torsion. We denote the ring of smooth functions on $M_n$ by $f\left(M_n\right)$, the Lie algebra of smooth vector fields on $M_n$ by $X\left(M_n\right)$ and arbitrary smooth vector fields on $M_n$ by $X,Y,Z$.

A diffeomorphism $f:A_n\to {\overline{A}}_n$ is called a geodesic mapping of $A_n$ onto ${\overline{A}}_n$ if f maps any geodesic curve on $A_n$ onto a geodesic curve on ${\overline{A}}_n$ [6,24,25,26,33].

A manifold $A_n$ admits a geodesic mapping onto ${\overline{A}}_n$ if and only if the equation [6,24,25,26,33]

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+\psi \left(X\right)Y+\psi \left(Y\right)X$$

holds for any vector fields $X,Y$ and where $\psi$ is a differential form on $M_n(={\overline{M}}_n)$.

If $\psi =0$ then geodesic mapping is called trivial and nontrivial if $\psi \ne 0$.

Let $V_n=(M_n,g)$ be an n-dimensional pseudo-Riemannian manifold with a metric tensor g and ∇ be a Levi-Civita connection.

A pseudo-Riemannian manifold $V_n$ admits a geodesic mapping onto a pseudo-Riemannian manifold ${\overline{V}}_n$ if and only if there exists a differential form $\psi$ on $V_n$ such that the Levi-Civita equation [6,24,26,33]

$$\left({\nabla }_Z\overline{g}\right)(X,Y)=2\psi \left(Z\right)\overline{g}(X,Y)+\psi \left(X\right)\overline{g}(Y,Z)+\psi \left(Y\right)\overline{g}(X,Z)$$

(1)

holds for any vector field $X,Y,Z$.

Or in the coordinate form

$${\overline{g}}_{ij,k}=2{\psi }_k{\overline{g}}_{ij}+{\psi }_i{\overline{g}}_{jk}+{\psi }_j{\overline{g}}_{ik},$$

(2)

where ${\psi }_i={\nabla }_i\mathsf{\Psi }$, $\mathsf{\Psi }$ is a scalar field, ${\overline{g}}_{ij}$ are components of the metric $\overline{g}$ and comma “ , ” denotes a covariant derivative with respect to ∇.

The Levi-Civita Equation $\left(1\right)$ is not linear so it is not convenient for investigations. Sinyukov [24,33] proved that a pseudo-Riemannian manifold $V_n$ admits a geodesic mapping if and only if there exist a differential form $\lambda$ and a regular symmetric bilinear form a on $V_n$ such that the equation

$$\left({\nabla }_{Z}a\right)(X,Y)=\lambda \left(X\right)g(Y,Z)+\lambda \left(Y\right)g(X,Z)$$

(3)

holds for any vector field $X,Y,Z$. Or in the coordinate form

$$a_{ij,k}={\lambda }_i{g}_{jk}+{\lambda }_j{g}_{ik},$$

(4)

where $a_{ij}$ and ${\lambda }_i$ are components of a and $\lambda$, respectively. Note that ${\lambda }_i={\nabla }_i\mathsf{\Lambda }$, $\mathsf{\Lambda }$ is a scalar field.

Solutions of $\left(2\right)$ and solutions of $\left(4\right)$ are related by the equalities

$$a_{ij}=exp\left(2\mathsf{\Psi }\left(x\right)\right)·{\overline{g}}^{\alpha \beta }{g}_{i\alpha }{g}_{j\beta }\mathrm{and}{\lambda }_i=-exp\left(2\mathsf{\Psi }\left(x\right)\right)·{\overline{g}}^{\alpha \beta }{g}_{i\alpha }{\psi }_{\beta }$$

where $g_{ij}$ are components of the metric g, $\left(g^{ij}\right)={\left(g_{ij}\right)}^{-1}$ and $\left({\overline{g}}^{ij}\right)={\left({\overline{g}}_{ij}\right)}^{-1}$.

If $V_n$$(n>2)$ admits two linearly independent solutions not proportional to the metric tensor g then [24]

$$\left({\nabla }_Y\lambda \right)\left(X\right)=Ka(X,Y)+\mu g(X,Y)\mathrm{and}{\nabla }_X\mu =2K\lambda \left(X\right),$$

(5)

where K is a constant and $\mu$ is a scalar field on $V_n$ or in the coordinate form

$${\nabla }_j{\lambda }_i=K{a}_{ij}+\mu g_{ij}\mathrm{and}{\nabla }_k\mu =2K{\lambda }_k.$$

(6)

A pseudo-Riemannian manifold satisfying the Equations (3) and (5) is called a $V_n\left(K\right)$-space.

These spaces for Riemannian manifolds were introduced by Solodovnikov [34] as $V\left(K\right)$-space and in another problem for pseudo-Riemannian manifolds were introduced by Mikeš [14,24] as $V_n\left(B\right)$-space (in this case $B=-K$).

A vector field $\phi$ on a pseudo-Riemannian manifold $V_n$ is called concircular if

$$\left({\nabla }_Y\phi \right)X=\varrho g(X,Y),$$

(7)

where $\varrho$ is a scalar field on $V_n$, see Reference [24] (p. 247), Reference [33] (p. 83) and Yano [38].

If $\varrho \ne 0$ a concircular field belongs to the basic type otherwise it belongs to the exceptional type.

A pseudo-Riemannian manifold $V_n$ admitting a concircular field is called an equidistant space [24,33]. The equidistant space belongs to the basic type if it admits a concircular field of the basic type and it belongs to the exceptional type if it admits concircular fields only of the exceptional type [33].

Concircular fields play an important role in the theories of conformal and geodesic mappings and transformations. They were studied by a number of geometers: Brinkmann [39], Fialkow [40], Yano [38], Sinyukov [33], Aminova [3], Mikeš [13,14,15,16,24], Shandra [28,29,30,31] and so forth.

Let us denote the linear space of all concircular fields on $V_n$ by $\mathrm{Con}\left(V_n\right)$. If $\stackrel{1}{\phi }\phantom{(},\dots ,\stackrel{m}{\phi }\phantom{(}$ is a basis in $\mathrm{Con}\left(V_n\right)$ then the tensor field

$$a=\sum _{\alpha ,\beta =1}^m\underset{\alpha \beta }{C}\phantom{(}(\stackrel{\alpha }{\phi }\otimes \stackrel{\beta }{\phi })$$

is a solution of the system (3), where $\underset{\alpha \beta }{C}\phantom{(}(=\underset{\beta \alpha }{C}\phantom{(})$ are some constants. So $V_n$ admits the geodesic mapping.

Pseudo-Riemannian manifolds admitting concircular fields form the class of manifolds which is closed with respect to the geodesic mappings [24,33]. Let a pseudo-Riemannian manifold $V_n$ admit a geodesic mapping onto a pseudo-Riemannian manifold ${\overline{V}}_n$, if there exists a concircular field $\phi$ on $V_n$ then there exists a concircular field $\overline{\phi }$ on ${\overline{V}}_n$ such that

$$\overline{\varrho }=exp\left(\mathsf{\Psi }\right)(\varrho +g^{ij}{\phi }_i{\psi }_j).$$

(8)

A concircular field $\phi$ is said to be special if

$$Z\left(\varrho \right)=Kg(Z,\phi ),$$

(9)

where K is a constant and it is said to be convergent if $\varrho$ is a constant. A pseudo-Riemannian manifold $V_n$ admitting a convergent field is called a Shirokov space, see References [24,31,32,33].

If there exist two linearly independent concircular fields on $V_n$ then all concircular fields on $V_n$ are special with the same constant K, see Reference [24]. A pseudo-Riemannian manifold $V_n$ admitting a special concircular field is a $V_n\left(K\right)$-space. On a $V_n\left(K\right)$-space any concircular field is special.

2. Shirokov Spaces and ${\mathit{V}}_{\mathit{n}}\left(\mathit{K}\right)$ Spaces $(\mathit{K}\ne \mathbf{0})$

Lemma 1.

Let a pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ admit convergent fields $\tilde{\phi }$ such that

$$a)\parallel \tilde{\phi }\parallel <0andb)\left({\tilde{\nabla }}_{\tilde{Y}}\tilde{\phi }\right)\tilde{X}=K·G(\tilde{X},\tilde{Y}),$$

(10)

for any vector field $\tilde{X},\tilde{Y}$ on $M_{n+1}$, where $K(\ne 0)$ is a constant. Then there exists the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ in which the components $G_{IJ}$ of the metric G are reduced to the form

$$G_{IJ}=exp\left(2K{x}^0\right)·\left(\begin{array}{cc}-1& 0\\ 0& {\displaystyle \frac{g_{ij}\left(x^k\right)}{K}}\end{array}\right)$$

(11)

where $g_{ij}\left(x^k\right)$ are the components of the metric of some $V_n=(M_n,g)$, $I,J,\dots =1,\dots ,n+1$, $i,j,\dots =1,\dots ,n$.

Proof. 

Let $\tilde{\phi }{}^I$ be the components of the vector fields $\tilde{\phi }$ g-conjugate with a convergent fields $\tilde{\phi }$ in a coordinate system $\left(x^I\right)$ on $V_{n+1}=(M_{n+1},G)$. Then due to (10b) they satisfy

$${\tilde{\nabla }}_J{\tilde{\phi }}^I=K{\delta }_J^I.$$

(12)

Let D be the linear space of all vector fields on $V_{n+1}$ which are orthogonal to $\stackrel{*}{\phi }$. It is easy to check that D is involutive. So if we use as a natural basis of $X\left(M_{n+1}\right)$ the basis $\left\{e_I\right\}=\{\stackrel{*}{\phi },e_i\}$, where $\left\{e_i\right\}$, is the basis in D, we get the coordinate system $\left(x^I\right)=(x^0,x^i)$ in which

$$\mathrm{a})\tilde{\phi }{}^I={\delta }_0^I;\mathrm{b})G_{i0}=0.$$

(13)

In these coordinates the Equations (12) are equivalent to

$${\tilde{\mathsf{\Gamma }}}_{0J}^I=K{\delta }_J^I,$$

(14)

where $\tilde{\mathsf{\Gamma }}{}_{JK}^I$ are the components of the Levi-Civita connection of the metric G.

Let us consider the conditions (14). If $I=0,J=j$ we have

$${\partial }_j{G}_{00}=0.$$

(15)

If $I=0,J=0$ we get

$${\partial }_j{G}_{00}=2K{G}_{00}.$$

(16)

It follows from (15) and (16) that $G_{00}=C·exp\left(2K{x}^0\right)$, where C is a constant. Due to (10a) it holds $C<0$. We can choose it such that $C=-1$. So

$$G_{00}=-exp\left(2K{x}^0\right).$$

(17)

If $I=i,J=j$ we obtain ${\partial }_0{G}_{ij}=2K{G}_{ij}$. So

$$G_{ij}=exp\left(2K{x}^0\right)\frac{g_{ij}\left(x^k\right)}{K}.$$

(18)

It follows from (13b), (17) and (18) that in the coordinate system $\left(x^I\right)=(x^0,x^i)$ the components $G_{IJ}$ reduce to the form (11).

Conversely, if the components $G_{IJ}$ of the metric G in the coordinate system $\left(x^I\right)=(x^0,x^i)$ reduce to the form (11) then the components ${\tilde{\mathsf{\Gamma }}}_{JK}^I$ of the Levi-Civita connection reduce to the form:

$${\tilde{\mathsf{\Gamma }}}_{00}^0=K,{\tilde{\mathsf{\Gamma }}}_{0j}^0=0,{\tilde{\mathsf{\Gamma }}}_{0j}^i={\delta }_j^i,{\tilde{\mathsf{\Gamma }}}_{ij}^0=g_{ij},{\tilde{\mathsf{\Gamma }}}_{ij}^k={\mathsf{\Gamma }}_{ij}^k,$$

(19)

where ${\mathsf{\Gamma }}_{ij}^k$ are the components of the Levi-Civita connection of the metric g. Using direct calculations it is easy to verify that a vector field with components $\tilde{\phi }{}_0^I={\delta }_0^I$ by virtue (19) satisfies the conditions (10a) and (12). □

Remark 1.

The components $G^{IJ}$ of the inverse metric G in the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ reduce to the form

$$G^{IJ}=exp(-2K{x}^0)\left(\begin{array}{cc}-1& 0\\ 0& K{g}^{ij}\left(x^k\right)\end{array}\right).$$

(20)

Lemma 2.

The pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ with the metric defined by the conditions (11) admits an absolutely parallel covector field $\tilde{\phi }$ if and only if its components in the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ reduce to the form

$${\tilde{\phi }}_I=exp\left(K{x}^0\right)\left(\varrho \left(x^k\right),{\phi }_i\left(x^k\right)\right),$$

(21)

where $\varrho \left(x^k\right)$ and ${\phi }_i\left(x^k\right)$ satisfy the following equations on $V_n=(M_n,g)$:

$${\nabla }_j{\phi }_i=\varrho g_{ij},$$

(22)

$${\nabla }_j\varrho =K{\phi }_j.$$

(23)

Proof. 

Let ${\tilde{\phi }}_I$ be the components of an absolutely parallel covector field $\tilde{\phi }$ in the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ on $V_{n+1}=(M_{n+1},G)$. So

$${\tilde{\nabla }}_J{\tilde{\phi }}_I=0$$

(24)

If $I=0,J=0$ we get from (24) by virtue (19): ${\partial }_0{\tilde{\phi }}_0-K{\tilde{\phi }}_0=0.$

Thus

$${\tilde{\phi }}_0=exp\left(K{x}^0\right)\varrho \left(x^k\right).$$

(25)

If $I=i,J=0$: ${\partial }_0{\tilde{\phi }}_i-K{\tilde{\phi }}_i=0.$ Hence,

$${\tilde{\phi }}_i=exp\left(K{x}^0\right){\tilde{\phi }}_i\left(x^k\right).$$

(26)

If $I=0,J=j$: ${\partial }_j{\tilde{\phi }}_0-K{\tilde{\phi }}_j=0.$ Due to (25) and (26) we have (23) and

if $I=i,J=j$: ${\partial }_j{\tilde{\phi }}_i-g_{ij}{\tilde{\phi }}_0-{\mathsf{\Gamma }}_{ij}^a{\tilde{\phi }}_a=0.$ Thus, we obtain (22).

Conversely, using direct calculations it is easy to check that if the covector field $\tilde{\phi }$ has components ${\tilde{\phi }}_i=exp\left(K{x}^0\right)(\varrho \left(x^k\right),{\phi }_i\left(x^k\right))$ in the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ on $V_{n+1}=(M_{n+1},G)$ with metric $\left(11\right)$, where $\varrho \left(x^k\right)$ and ${\phi }_i\left(x^k\right)$ satisfy the Equations (22) and (23) on $V_n=(M_n,g)$, then $\tilde{\phi }$ due to (19) it is absolutely parallel. □

Remark 2.

The Equations (22) and (23) are the coordinate forms of the Equations (7) and (9) defining a special concircular field. So the conditions $\left(21\right)$ establish a one-to-one correspondence between absolutely parallel covector fields on the Shirokov space $V_{n+1}=(M_{n+1},G)$ and special concircular fields on the $V_n\left(K\right)$-space $K\ne 0$.

In a similar way, it is possible to prove the following statement.

Lemma 3.

The pseudo-Riemannian manifold $V_{n+1}=(M_{n+1},G)$ with the metric defined by the conditions $\left(11\right)$ admits an absolutely parallel symmetric bilinear form $\tilde{a}$ if and only if its components in the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ reduce to the form

$${\tilde{a}}_{IJ}=exp\left(2K{x}^0\right)\left(\begin{array}{cc}\mu \left(x^k\right)& {\lambda }_i\left(x^k\right)\\ {\lambda }_j\left(x^k\right)& a_{ij}\left(x^k\right)\end{array}\right)$$

(27)

where $a_{ij}\left(x^k\right)$, ${\lambda }_i\left(x^k\right)$ and $\mu \left(x^k\right)$ satisfy the Equations $\left(4\right)$ and $\left(6\right)$ on $V_n=(M_n,g)$.

Remark 3.

The Equations $\left(4\right)$ and $\left(6\right)$ define a $V_n\left(K\right)$-space. So the conditions $\left(27\right)$ establish a one-to-one correspondence between absolutely parallel symmetric bilinear forms on the Shirokov space $V_{n+1}=(M_{n+1},G)$ and solutions of the system $\left(4\right)$ and $\left(6\right)$ defining geodesic mappings of the $V_n\left(K\right)$-space $(K\ne 0)$.

Remark 4.

The set of absolutely parallel symmetric bilinear forms on $V_n=(M_n,g)$ is a special Jordan algebra $J_0$ with the operation of multiplication $\stackrel{1}{A}\ast \stackrel{2}{A}=\{\stackrel{1}{A};\stackrel{2}{A}\}$, where A is the linear operator g-conjugate with a bilinear form a, defined by $g(AX,Y)=a(X,Y)$ and $\{\stackrel{1}{A};\stackrel{2}{A}\}$ are Jordan brackets

$$\{\stackrel{1}{A};\stackrel{2}{A}\}=\frac{1}{2}\left(\stackrel{1}{A}\stackrel{2}{A}+\stackrel{2}{A}\stackrel{1}{A}\right).$$

(28)

The condition $\left(28\right)$ can be rewritten in the vector form as

$$2\{\stackrel{1}{a};\stackrel{2}{a}\}(X,Y)=\stackrel{1}{a}\left(\stackrel{2}{A}X,Y\right)+\stackrel{1}{a}\left(\stackrel{2}{A}Y,X\right)$$

(29)

or in the coordinate form

$$2{\{\stackrel{1}{a};\stackrel{2}{a}\}}_{ij}=g^{ab}\left(\stackrel{1}{a}{}_{ai}\stackrel{2}{a}{}_{bj}+\stackrel{1}{a}{}_{aj}\stackrel{2}{a}{}_{bi}\right).$$

(30)

This statement follows from the Lemma 2.

Theorem 1.

The set of solutions of the system $\left(4\right)$ and $\left(6\right)$ on a $V_n\left(K\right)$-space $(K\ne 0)$ forms a special Jordan algebra J with the operation of multiplication $\left\{(\stackrel{1}{a},\stackrel{1}{\lambda },\stackrel{1}{\mu });(\stackrel{2}{a},\stackrel{2}{\lambda },\stackrel{2}{\mu })\right\}=(\stackrel{3}{a},\stackrel{3}{\lambda },\stackrel{3}{\mu })$, where

$$2\stackrel{3}{a}(X,Y)=K\left(\stackrel{1}{a}(\stackrel{2}{A}X,Y)+\stackrel{1}{a}(\stackrel{2}{A}Y,X)\right)-\left(\stackrel{1}{\lambda }\otimes \stackrel{2}{\lambda }+\stackrel{2}{\lambda }\otimes \stackrel{1}{\lambda }\right)(X,Y),$$

(31)

$$2\stackrel{3}{\lambda }\left(X\right)=K\left(\stackrel{1}{\lambda }(\stackrel{2}{A}X)+\stackrel{2}{\lambda }(\stackrel{1}{A}X)\right)-\left(\stackrel{1}{\mu }\stackrel{2}{\lambda }(X)+\stackrel{2}{\mu }\stackrel{1}{\lambda }(X)\right),$$

(32)

$$\stackrel{3}{\mu }=K{g}^{-1}\left(\stackrel{1}{\lambda }\stackrel{2}{\lambda }\right)-\stackrel{1}{\mu }\stackrel{2}{\mu }.$$

(33)

The algebra J is isomorphic to the special Jordan algebra $J_0$ of absolutely parallel symmetric bilinear forms on the Shirokov space $V_{n+1}=(M_{n+1},G)$ with the metric $\left(11\right)$.

Proof of the theorem follows immediately from the Lemma 2 and $\left(20\right)$, $\left(27\right)$ and $\left(30\right)$.

Remark 5.

Due to $\left(29\right)$ the unit of the algebra $J_0$ is G so the unit of the algebra J is ${\displaystyle \left(\frac{g}{K},0,-1\right)}$.

Remark 6.

If there exists a convergent field $\tilde{\phi }$ on $V_{n+1}=(M_{n+1},G)$ such that $\parallel \tilde{\phi }\parallel >0$, then there exists the adapted coordinate system $\left(x^I\right)=(x^0,x^i)$ in which the components $G_{IJ}$ of the metric G reduce to the form

$$G_{IJ}=exp\left(2K{x}^0\right)\left(\begin{array}{cc}1& 0\\ 0& {\displaystyle \frac{-g_{ij}\left(x^k\right)}{K}}\end{array}\right),$$

where $g_{ij}\left(x^k\right)$ are the components of the metric of some $V_n=(M_n,g)$. Using this metric and $\left(29\right)$ we can define a new operation of multiplication ${\{·,·\}}_2$. It is obvious that $\{\stackrel{1}{A};\stackrel{2}{A}\}=-{\{\stackrel{1}{A}\phantom{(};\stackrel{2}{A}\phantom{(}\}}_2$.

Corollary 1.

Let $V_n=(M_n,g)$ be a $V_n\left(K\right)$-space $(K\ne 0)$ then there exists the solution $(a,\lambda ,\mu )$ of the system $\left(4\right)$ and $\left(6\right)$ satisfying the following conditions:

$$Ka(AX,Y)-(\lambda \otimes \lambda )(X,Y)=\frac{eg(X,Y)}{K},$$

(34)

$$K\lambda \left(AX\right)-\mu \lambda \left(X\right)=0,$$

(35)

$$K{g}^{-1}(\lambda ,\lambda )-{\mu }^2=-e,$$

(36)

where e takes values $\pm 1,0$.

Proof. 

Let $\tilde{b}$ be an absolutely parallel symmetric bilinear form on the Shirokov space $V_{n+1}=(M_{n+1},G)$ with the metric $\left(11\right)$. Then as it has been shown in Reference [11] there exists the absolutely parallel symmetric bilinear form $\tilde{a}$ on $V_{n+1}=(M_{n+1},G)$ such that $\tilde{A}{}^2=e$ or in the equivalent form

$$\tilde{a}(\tilde{A}\tilde{X},\tilde{Y})=eG(\tilde{X},\tilde{Y}).$$

(37)

The Equation $\left(37\right)$ means that $\{\tilde{a},\tilde{a}\}=eG$. Hence if $(a,\lambda ,\mu )$ is the corresponding solution of the system $\left(4\right)$ and $\left(6\right)$ on the $V_n\left(K\right)$-space $(K\ne 0)$ then taking into account $\left(31\right)$–$\left(33\right)$ we get $\left(34\right)$–$\left(36\right)$. □

As mentioned above concircular fields generate a solution of the Equation $\left(2\right)$. Denote this set of solutions by $J_c$.

Theorem 2.

$J_c$ is an ideal of J.

Proof. 

To prove that $J_c$ is an ideal of J on $V_n=(M_n,g)$ it is equivalent to prove that $J_{0c}$ is an ideal of $J_0$ on $V_{n+1}=(M_{n+1},G)$, where $J_{0c}$ is the set of absolutely parallel symmetric bilinear forms generated by absolutely parallel covector fields.

Let $\stackrel{1}{\phi },\dots ,\stackrel{m}{\phi }$ be a basis of the linear space Conv($V_{n+1}$) of absolutely parallel covector fields on $V_{n+1}=(M_{n+1},G)$. Then any absolutely parallel symmetric bilinear form generated by absolutely parallel covector fields has the components

$${\tilde{b}}_{IJ}=\sum _{\alpha ,\beta =1}^m\underset{\alpha \beta }{C}(\stackrel{\alpha }{\phi }{}_I\stackrel{\beta }{\phi }{}_J),$$

where $\underset{\alpha \beta }{C}(=\underset{\beta \alpha }{C})$ are some constants. Let ${\tilde{a}}_{IJ}$ be the components of the arbitrary absolutely parallel symmetric bilinear form $\tilde{a}$. We should prove that $\{\tilde{a},\tilde{b}\}\in J_{0c}$. We have

$$2\{\tilde{a},\tilde{b}\}=G^{DT}\sum _{\alpha ,\beta =1}^m\underset{\alpha \beta }{C}(\stackrel{\alpha }{\phi }{}_I\stackrel{\beta }{\phi }{}_D{\tilde{a}}_{TJ}+\stackrel{\alpha }{\phi }{}_J\stackrel{\beta }{\phi }{}_D{\tilde{a}}_{TI})=\sum _{\alpha ,\beta =1}^m\underset{\alpha \beta }{C}(\stackrel{\alpha }{\phi }{}_I\stackrel{\beta }{\mathsf{\Phi }}{}_J+\stackrel{\alpha }{\phi }{}_J\stackrel{\beta }{\mathsf{\Phi }}{}_I),$$

(38)

where $\stackrel{\beta }{\mathsf{\Phi }}{}_I=\stackrel{\beta }{\phi }{}_D{\tilde{a}}_{TI}{G}^{DT}$ is an absolutely parallel covector field. Therefore,

$$\stackrel{\beta }{\mathsf{\Phi }}{}_I=\sum _{\gamma =1}^m{F}_{\gamma }^{\beta }\stackrel{\gamma }{\phi }{}_I$$

(39)

where $F_{\gamma }^{\beta }$ are some constants. It follows from $\left(38\right)$ and $\left(39\right)$ that

$$2{\{\tilde{a},\tilde{b}\}}_{IJ}=\sum _{\alpha ,\beta ,\gamma =1}^m\left(F_{\beta }^{\gamma }\underset{\alpha \gamma }{C}+F_{\alpha }^{\gamma }\underset{\beta \gamma }{C}\right)\stackrel{\alpha }{\phi }{}_I\stackrel{\beta }{\phi }{}_J.$$

Thus, $\{\tilde{a},\tilde{b}\}\in J_{0c}$. □

3. ${\mathit{V}}_{\mathit{n}}\left(\mathbf{0}\right)$-Spaces

Let $(M_n.g)$ be a $V_n\left(0\right)$-space, then there exists a solution of the system

$${\nabla }_k{a}_{ij}={\lambda }_i{g}_{jk}+{\lambda }_j{g}_{ik},$$

(40)

$${\nabla }_k{\lambda }_i=\mu g_{ik},$$

(41)

where $\mu$ is a constant and ${\lambda }_i={\nabla }_i\mathsf{\Lambda }$. Thus, a $V_n\left(0\right)$-space is a Shirokov space.

Lemma 4.

If the $V_n\left(0\right)$-space does not admit any convergent field of the basic type and φ is an absolutely parallel covector field on it, then there exists the sequence of absolutely parallel covector fields $\left\{\stackrel{\alpha }{\phi }\right\}(\alpha \in \mathbb{N})$ such that

$$a)\stackrel{\alpha +1}{\phi }\left(X\right)=\stackrel{\alpha }{\phi }\left(AX\right)-\stackrel{\alpha }{f}\lambda \left(X\right),b)\stackrel{\alpha }{\phi }\left({\lambda }^{*}\right)=0,\forall \alpha \in \mathbb{N},$$

(42)

where $\stackrel{1}{\phi }=\phi$, $d\stackrel{\alpha }{f}=\stackrel{\alpha }{\phi }$, ${\lambda }^{*}$ is the vector field g-conjugate with λ.

Proof. 

Taking into account that the $V_n\left(0\right)$ does not admit any convergent fields of the basic type we obtain from $\left(41\right)$ that

$${\nabla }_k{\lambda }_i=0.$$

(43)

Let ${\phi }_i$ be the components of an absolutely parallel covector field $\phi$ on a $V_n\left(0\right)$. Denote $\stackrel{1}{\phi }=\phi$. Consider the covector field

$$\stackrel{2}{\phi }{}_i=a_i^t\stackrel{1}{\phi }-\stackrel{1}{f}{\lambda }_i,$$

(44)

where $a_i^t$ are components of the linear operator $A(a_i^j=g^{jl}{a}_{il})$. It follows from $\left(44\right)$ due to $\left(40\right)$ and $\left(43\right)$

$${\nabla }_k\stackrel{2}{\phi }{}_i=\stackrel{1}{\phi }{}_t{\lambda }^t{g}_{ik},$$

(45)

where ${\lambda }^t=g^{ti}{\lambda }_i$. According to our assumption it follows from $\left(45\right)$ that

$$\stackrel{1}{\phi }{}_t{\lambda }^t=0\mathrm{and}{\nabla }_k\stackrel{2}{\phi }{}_i=0.$$

Applying now similar argumentation to the covector $\stackrel{2}{\phi }{}_i$ and continuing the process in this way, we obtain the desired sequence. □

Remark 7.

The Equation (42b) due to (42a) can be rewritten as

$$\phi (\stackrel{\alpha -1}{\lambda }{}^{*})=0,\forall \alpha \in \mathbb{N},$$

(46)

where $\stackrel{\alpha }{A}$ is the α-s power of the linear operator A.

Theorem 3.

Let a pseudo-Riemannian manifold $V_n$ be a $V_n\left(0\right)$-space. Then there exists a convergent field of the basic type on $V_n$ or there exists the sequence of linearly independent absolutely parallel covector fields $\{\stackrel{\alpha }{\lambda }\}$, $(\alpha =1,2\dots ,p\le n-1)$ such that

$$\stackrel{\alpha +1}{\lambda }\left(X\right)=\stackrel{\alpha }{\lambda }\left(AX\right)-\stackrel{\alpha }{\mathsf{\Lambda }}\alpha \left(X\right),\lambda (\stackrel{\alpha -1}{A}{\lambda }^{*})=0,\forall \alpha \in A,$$

(47)

$$\stackrel{p}{\lambda }(AX)=\stackrel{p}{\mathsf{\Lambda }}\lambda (X),$$

(48)

where $\stackrel{1}{\lambda }=\lambda$, ${\lambda }^{*}$ is the vector field g-conjugate with λ.

Proof. 

(1) It follows from $\left(41\right)$ that if $\mu \ne 0$ then $\lambda$ is a convergent field of the basic type on $V_n\left(0\right)$.

(2) Let $\mu =0$, then $\nabla \lambda =0$. According to the Lemma 4 and the Remark 7 we can construct the sequence of absolutely parallel covector fields $\{\stackrel{\alpha }{\lambda }\}(\alpha \in \mathbb{N})$ such that

$$\stackrel{\alpha +1}{\lambda }\left(X\right)=\stackrel{\alpha }{\lambda }\left(AX\right)-\stackrel{\alpha }{\mathsf{\Lambda }}\lambda \left(X\right),\lambda (\stackrel{\alpha -1}{A}{\lambda }^{*})=0,\forall \alpha \in \mathbb{N}.$$

This sequence contains no more than $p(\le n-1)$ linearly independent covectors. Otherwise, $V_n\left(0\right)$ will be locally flat and so it will admit a convergent field of the basic type. Thus,

$$\stackrel{p+1}{\lambda }=\sum _{\alpha =1}^p{C}_{\alpha }\stackrel{\alpha }{\lambda },$$

where $C_{\alpha }$ are constants and $\stackrel{1}{\lambda },\dots ,\stackrel{p}{\lambda }$ are linearly independent. Changing $\stackrel{\alpha }{\mathsf{\Lambda }}$ (defined to a constant) we can make $\stackrel{p+1}{\lambda }=0$. So we get $\left(48\right)$. □

Corollary 2.

If the $V_n\left(0\right)$-space does not admit any converging fields of the basic type and φ is an absolutely parallel covector field on it, then

$$\stackrel{\alpha -1}{\lambda }({\phi }^{*})=0,\forall \alpha \in \mathbb{N}$$

(49)

where ${\phi }^{*}$ is the vector field g-conjugate with φ.

Proof. 

We get from $\left(46\right)$: $(\stackrel{\alpha -1}{A}{\lambda }^{*})=\stackrel{\alpha -1}{A}\lambda \left({\phi }^{*}\right)=\stackrel{\alpha -1}{\lambda }\left({\phi }^{*}\right)=0$. □

The following statement holds.

Theorem 4.

Let a pseudo-Riemannian manifold $V_n$ admit a geodesic mapping onto a pseudo-Riemannian manifold ${\overline{V}}_n$ if there exists a concircular field of the basic type on ${\overline{V}}_n$, then there exists a concircular field of the basic type on $V_n$.

Proof. 

Let $\overline{\phi }$ be a concircular field of the basic type on ${\overline{V}}_n$ ($\overline{\varrho }\ne 0$), then there exists a concircular field $\phi$ on $V_n$. Let us suppose the contrary, namely that $V_n$ does not admit concircular fields of the basic type. It means that $\varrho =0$. So $\phi$ is an absolutely parallel covector field and, therefore, $V_n$ is a $V_n\left(0\right)$-space [30]. So according to Theorem 3 there exists a $V_n$ on the sequence of linearly independent absolutely parallel covector fields $\{\stackrel{\alpha }{\lambda }\}(\alpha =1,2,\dots ,p\le n-1)$ satisfying $\left(47\right)$ and $\left(48\right)$. The Equation $\left(48\right)$ in the coordinate form can be written as

$$a_i^t\stackrel{p}{\lambda }{}_t=\stackrel{p}{\mathsf{\Lambda }}{\lambda }_i.$$

(50)

Contracting $\left(50\right)$ with $\overline{a}{}_j^i$ (the inverse operator to $a_j^i$) by i and taking into account that ${\lambda }_i=-a_i^t{\psi }_t$ we get

$$\stackrel{p}{\lambda }{}_j=-\stackrel{p}{\mathsf{\Lambda }}{\psi }_j.$$

(51)

The condition $\left(49\right)$ means that ${\phi }^t\stackrel{p}{\lambda }{}_t=0$. Hence, due to $\stackrel{p}{\mathsf{\Lambda }}\ne 0$ it follows from $\left(51\right)$ that ${\phi }^t{\psi }_t=0.$ On the other hand since $\overline{\varrho }\ne 0$ and $\varrho =0$ the Equation $\left(8\right)$ gives us ${\phi }^t{\psi }_t\ne 0.$ This contradiction proves the theorem. □

Remark 8.

The Theorem 4 shows that pseudo-Riemannian manifolds admitting a concircular field of the basic type (i.e., equidistant spaces of the basic type) form the class of manifolds closed with respect to the geodesic mappings. The same properties have spaces of constant curvature [24,33], Einstein spaces [17,24] and $V_n\left(K\right)$-spaces [24].

Corollary 3.

Let an equidistant space of the exeptional type $V_n$ admit a geodesic mapping onto a pseudo-Riemannian manifold ${\overline{V}}_n$, then ${\overline{V}}_n$ is an equidistant space of the exeptional type.