1. Introduction
The problem of geodesic mappings of Riemannian manifolds was first introduced by T. Levi-Civita in the study of problems in mechanics [
1]. There are many monographs and papers devoted to the theory of geodesic mappings and transformations, their generalizations, and applications [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20]. In addition, A. Z. Petrov [
11] used geodesic mappings and their generalizations of pseudo-Riemannian spaces for models of gravitation fields. The above-mentioned spaces that generalize semi-Riemannian spaces with degenerate metrics are found in various applications, in particular unified field theories. As it was shown in [
14], in the case when the torsion tensor is semisymmetric, setting the Levi-Civita pseudo-connection is equivalent to setting the Weyl connection used in a unified field theory combining gravity and electromagnetism. Linear idempotent operators are used to define calibration fields that define different types of interactions. The theory of the multidimensional Universe uses degenerate Kaluza–Klein metrics [
21,
22].
The basic equations of geodesic mappings for pseudo-Riemannian manifolds were obtained by Levi-Civita, but they were non-linear [
8,
14,
16,
18]. The basic equations of geodesic mappings for pseudo-Riemannian manifolds in linear form were obtained by N. S. Sinyukov [
16]. These equations greatly advanced the study of geodesic maps and allowed us to obtain many interesting results. In particular, it reduced the question of whether a given pseudo-Riemannian manifold admits a non-trivial geodesic mapping to the analysis of a system of linear algebraic equations.
Analogues of the Sinyukov equations for holomorphic-projective mappings of Kähler manifolds were obtained by J. Mikes [
8]. However, all existing generalizations of geodesic mappings assume that the metric tensor of a pseudo-Riemannian manifold is nondegenerate. However, in physics and mechanics, there are models in which the metric tensor is degenerate [
15].
In this paper, we generalize the results of geodesic mappings of pseudo-Riemannian manifolds to the case of semi-Riemannian spaces with a degenerate metric. In particular, we will obtain analogues of the Levi-Civita equations and the Sinyukov equations. For our research, we use the theory of idempotent pseudo-connections [
15].
2. Preliminaries
Let be a smooth n-dimensional manifold. We denote the ring of smooth functions on by , the Lie algebra of smooth vector fields on by , and arbitrary smooth vector fields on by , and W.
Definition 1. A linear pseudo-connection onis a pair of operators, where:and h is a linear operator on, which forsatisfies the following conditions [
14]:
In the case where , any linear pseudo-connection is a linear connection on .
Definition 2. The torsion and curvature tensors of the linear pseudo-connectionare defined as follows [
14]:
Definition 3. A linear pseudo-connectionis said to be idempotent if it satisfies the following conditions [
14]:
In this case, h is called the horizontal projector, andis called the vertical projector. Here,means.
The torsion and curvature tensors of an idempotent pseudo-connection satisfy the following conditions [
14]:
Definition 4. A linear pseudo-connectionis said to be completely idempotent if it satisfies the following conditions [
14]:
where means . A manifold on which is given a completely idempotent pseudo-connection
with
is denoted by
. The completely idempotent pseudo-connection is an idempotent pseudo-connection [
14].
The torsion and curvature tensors of a completely idempotent pseudo-connection satisfy the following conditions [
14]:
Definition 5. A pair, where h is a linear operator and g is a bilinear form, is called an HR-structure of rank r if they satisfy the following conditions [
14]:
A manifoldwith an-structure is called a semi-Riemannian manifold and is denoted by.
For any
-structure
, there is a unique linear pseudo-connection
, called the
Levi-Civita pseudo-connection, that satisfies the conditions [
14]
It is defined by the formula [
14]
3. Geodesic Mappings of Manifolds with an Idempotent Pseudo-Connection
Let be an n-dimensional manifold with an idempotent pseudo-connection .
Definition 6. A curveonis called a geodesic if it satisfies the following condition:where X is a tangent vector of, andis a function of parameter t. Let
be components of the pseudo-connection
, and
be components of the tangent vector
X in some coordinate system on
. Then, Equation (
10) can be written in the equivalent form
We remark that a curve
on
with an affine connection
is called an
F-planar curve if it satisfies [
8]
where
F is a linear operator, and
and
are some functions of
t.
If
F is an almost product structure (
), then
are
horizontal and
vertical projectors, respectively. Then,
It follows from (
9) that the curve
is the geodesic curve with respect to the pseudo-connection
and from (
10) that the curve
is the geodesic with respect to the pseudo-connection
.
Definition 7. A diffeomorphism f:is called a geodesic mapping ofontoif f maps any geodesic ononto a geodesic on.
Theorem 1. A manifold with an idempotent pseudo-connection admits a geodesic mapping onto a manifold with the idempotent pseudo-connection if and only if the equationholds for any vector fields , where ψ is a differential form on , and the tensor satisfies the following conditions: Proof. Let
f:
be a geodesic mapping. Therefore, a geodesic
on
maps onto a geodesic
on
. Then, in a common coordinate system
with respect the mapping
f, the curve
satisfies (
11), and
satisfies the following conditions:
Subtracting Equation (
11) from this equation, we obtain
Multiplying the above formula by
, and alternating by
i and
l, we obtain
where
The relations (
15) are fulfilled identically with respect to
X, so it follows from (
15) that
Thus, contracting (
16) in
j and
m, we obtain
It follows from (
17) that
where
Thus, we have found the symmetric part of the deformation tensor
The conditions (
18)–(
20) are equivalent to (
12)–(
14). Conversely, it is easy to check that if the conditions (
12)–(
14) hold, then any geodesic on
will be a geodesic on
. □
Theorem 2. Then, a manifold with a completely idempotent pseudo-connection admits a geodesic mapping onto a manifold with a completely idempotent pseudo-connection if and only if the equationholds for any vector fields , where ψ is a differential form on . Proof. Taking into account (
5), we obtain
Thus, we have from (
23) and (
24)
It follows from (
25) that
In addition, according to (
21), we obtain
We obtain from (
29), due to (
26)–(
28),
Substituting (
30) into (
12), we find
or
The theorem is proved. □
Definition 8. If, then geodesic mapping is called trivial, and nontrivial if.
Definition 9. A geodesic mapping of a manifoldwith a completely idempotent pseudo-connectiononto a manifoldwith a completely idempotent pseudo-connectionis called canonical if Corollary 1. A manifold with a completely idempotent pseudo-connection admits a canonical geodesic mapping onto a manifold with a completely idempotent pseudo-connection if and only if in the Equation (
22),
Proof. The condition (
32) is equivalent to
Thus, if
, then we have from (
25) that
. Conversely, if
, then we obtain from (
30) that
. The corollary is proved. □
It follows from (
22) that the equation of a canonical geodesic mapping of manifolds with a completely idempotent pseudo-connection due to (
33) is equivalent to the equation
The Equations (
34) and (
35) can be rewritten in the coordinate form as
and these equations are the generalization of the equations of geodesic mappings of manifolds with an affine connection [
8,
16].
4. Completely Canonical Geodesic Mappings of Semi-Riemannian Manifolds
Let be a semi-Riemannian manifold with an -structure and ∇ be a Levi-Civita pseudo-connection.
Theorem 3. A semi-Riemannian manifold admits a canonical geodesic mapping onto a semi-Riemannian manifold if and only if there exists a differential form on such that equationshold for any vector fields . The validity of this statement follows from (
8), (
32), (
34), and (
35).
The coordinate form of Equations (
38) and (
39) can be given by the following formulas:
Equations (
35), (
38), and (
39) are the generalization of the equations of geodesic mappings of pseudo-Riemannian manifolds [
8,
16].
Definition 10. A canonical geodesic mapping of a semi-Riemannian manifoldonto a semi-Riemannian manifoldis called completely canonical if there exists a functionsuch that in Equation (38) satisfies,
and in the coordinate form This shows that is a gradient covector.
Theorem 4. If the affinor h of the -structure is integrable then any canonical geodesic mapping of a semi-Riemannian manifold is completely canonical.
Proof. If the affinor
h of the
-structure
is integrable, then there exists the adapted coordinate system
on
that the components of
h reduce to the form
where
follow from 1 to
r, and
follow from
to
n. It follows from (
6), (
7), and (
43) that in this coordinate system,
where
is the semi-inverse matrix to
; thus,
and
is the inverse matrix to
Contracting (
36) in
i and
j, we obtain
It follows from (
9) and (
44)–(
46) that
or
The theorem is proved. □
Theorem 5. A semi-Riemannian manifold admits a completely canonical geodesic mapping if and only if there exist a differential form and a bilinear form on such that the equationshold for any vector fields . Proof. Let
be the components of a semi-inverse tensor to
; thus,
Then, it follows from (
40) and (
41) by virtue of (
52) that
where
.
It easy to find from (
54), due to (
53), the following equations:
where
Equations (
55)–(
59) are equivalent to (
47)–(
51).
Conversely, if there exist a differential form
and a bilinear form
on
such that Equations (
55)–(
59) hold, then there exist an
HR-structure
and a differential form
such that Equations (
37)–(
39) and (
42) hold, where
The theorem is proved. □
Contracting (
55) by
in
i and
j, we obtain
Thus,
is a gradient covector.
It follows from (
60) and (
61) that
if and only if
.
Equations (
47)–(
51) generalize N. S. Sinyukov’s equations for geodesic mappings of pseudo-Riemannian manifolds [
8,
16].
Equation (
55) can be rewritten in the equivalent form
The integrability conditions of Equation (
55) on the basis of (
62), (
63), and the Ricci identities take the following form [
14]:
where
are components of the curvature tensor
R:
Contracting (
64) by
in
j and
k, and (
66) by
in
i and
j, we find
Thus, we obtain from (
67) and (
68)
where
is a certain scalar field.
Similarly, analysing the integrability conditions of Equation (
69) on the basis of (
67) and (
68), it is not difficult to obtain equations that
satisfies:
where
is the Ricci tensor, and
is the nonholonomy tensor of the horizontal distribution
.
Thus, the following theorem is proved.
Theorem 6. In order that a semi-Riemannian manifold admit a completely canonical geodesic mapping, it is necessary and sufficient that the system (55)–(59), (69), and (70) has a solution . Theorem 6 is a generalization of the main theorem of geodesic mappings of pseudo-Riemannian manifolds. The system of Equations (
55), (
69), and (
70) forms a closed system of first-order linear partial differential equations of the Cauchy type. The integrability conditions of these equations, as well as their differential prolongations, will also be linear. Thus, the question of whether a given semi-Riemannian manifold
admits a completely canonical geodesic mapping is reduced to the analysis of the consistency of a certain system of linear algebraic equations.
5. Completely Canonical Geodesic Mappings and Concircular Fields
Definition 11. A vector fieldon a semi-Riemannian manifoldsatisfying the conditions [
13]
where is a scalar field on , is called a concircular field on . Here, we mean the covariant derivative of the covector field. A covariant derivative with respect to the pseudo-connection can be defined for a tensor field of any type. You can read about this in [14]. The Equations (
71) and (
72) can be rewritten in the equivalent coordinate form
If , a concircular field belongs to the main type, and it belongs to the exceptional type otherwise.
Theorem 7. Let φ be a concircular field on a semi-Riemannian manifold . If admits a nontrivial completely canonical geodesic mapping onto a semi-Riemannian manifold , then there exists a concircular field on .
Proof. Let
be the components of the concircular field
on
. Then,
is the components of the concircular field
on
due to (
36) and
The theorem is proved. □
Theorem 8. Let φ be a concircular field of the main type on a semi-Riemannian manifold ; then, admits a nontrivial completely canonical geodesic mapping.
Proof. Let
be the components of the concircular field
on
. Then, the tensor
satisfies (
55)–(
57) and (
59), where the constant
C is chosen in a way that
and where
It follows from the integrability conditions of Equation (
73) that
Whereas, for the Levi-Civita pseudo-connection
it follows from (
75) and (
76) that tensor
satisfies (
58). Thus,
satisfies (
55)–(
59), and according to Theorem 5, the space
admits a nontrivial completely canonical geodesic mapping. The theorem is proved. □
6. Conclusions
In this paper, we study geodesic mappings of manifolds with idempotent pseudo-connections. We obtained the basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric. We proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings.