Geodesic Mappings Research Articles

On canonical almost geodesic mappings which preserve the weyl projective tensor

We study a partial case of canonical almost geodesic mappings of the first type of spaces with affine connection that preserve Weyl projective curvature tensor and certain other tensors. Main equations under consideration are reduced to a closed Cauchy system type in covariant derivatives. Therefore a general solution to these equations depends on a finite number of constants. We submit an example of above mappings between flat spaces. We establish that projective Euclidean and equiaffine spaces form closed classes of spaces with respect to these mappings.

On the problem of geodesic mappings and deformations of generalized Riemannian spaces

This paper is devoted to a study of geodesic mappings and infinitesimal geodesic deformations of generalized Riemannian spaces. While a geodesic mapping between two generalized Riemannian spaces any geodesic line of one space sends to a geodesic line of the other space, under an infinitesimal geodesic deformation any geodesic line is mapped to a curve approximating a geodesic with a given precision. Basic equations of the theory of geodesic mappings in the case of generalized Riemannian spaces are obtained in this paper. A new generalization of the famous Levi Civita's equation is found. Necessary and sufficient conditions for a nontrivial infinitesimal geodesic deformation are given. It is proven that a generalized Riemannian space admits nontrivial infinitesimal geodesic deformations if and only if it admits nontrivial geodesic mappings. At last it is shown that generalized equidistant spaces of primary type admit nontrivial geodesic deformations.

Geodesic mapping onto Kählerian space of the third kind

In the present paper we study geodesic mappings between generalized Riemannian GRN and generalized Kählerian spaces of the third type GK3N, and specially the case when these spaces have the same torsions at corresponding points. Using the non-symmetric metric tensor we find necessary and sufficient conditions for the existence of geodesic mapping f:GRN→GK3‾N with respect to the four kinds of covariant derivatives.

Geodesic Mappings Onto Riemannian Manifolds and Differentiability

Geodesic Mappings Onto Riemannian Manifolds and Differentiability

Geodesic Mappings of Spaces with Affine Connection Onto Symmetric Manifolds

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto symmetric manifolds. We obtain fundamental equations of this problem. At the end of our paper we demonstrate example of studied mappings.

Geodesic Mappings and Their Generalizations

This paper is devoted to further study of the theory of geodesic mappings and their generalizations, including conformal, holomorphically projective, F-planar, and almost geodesic mappings of affinely connected spaces.

Some Invariants of Equitorsion Third Type Almost Geodesic Mappings

In this paper, we consider equitorsion third type almost geodesic mappings of non-symmetric affine connection spaces. Applying different computational methods, we find invariants of these mappings obtained from curvature tensors of a corresponding non-symmetric affine connection space. Weyl projective tensor of a symmetric affine connection space is generalized in this way for the case of a third type almost geodesic mapping.

Necessary and sufficient conditions for equitorsion geodesic mapping

In the present paper we study equitorsion geodesic mappings between two generalized Riemannian spaces f:GRN→GR‾N. In this case these spaces have the same torsions at corresponding points. We prove that a generalized Riemannian space GRN admits an equitorsion geodesic mapping onto a generalized Riemannian space GR‾N if and only if in GRN certain differential equations of Cauchy type in covariant derivatives of the θ=1,…,4 kinds have a solution with respect to the unknown tensor aij, the gradient vector λi≠0, and the differentiable function μθ, θ=1,…,4. In fact we find four systems of PDE all equivalent to the existence of an equitorsion geodesic mapping and discuss the number of linearly independent solutions of this system of PDE. We establish an upper bound for the number of solutions for the geometrical problem under consideration.

Basic equations of G-almost geodesic mappings of the second type, which have the property of reciprocity

We study G-almost geodesic mappings of the second type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2 between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider e-structures that generate mappings of type $$\mathop {{\pi _2}}\limits_\theta (e),\theta = 1,2$$ , θ = 1,2. For a mapping $$\mathop {{\pi _2}}\limits_\theta (e,F),\theta = 1,2$$ , θ = 1,2, we determine the basic equations which generate them.

On special first-type almost geodesic mappings of affine connection spaces preserving a certain tensor

On special first-type almost geodesic mappings of affine connection spaces preserving a certain tensor

Special Almost Geodesic Mappings of the First Type of Non-symmetric Affine Connection Spaces

We introduce a special kind of almost geodesic mappings of the first type \(\pi _1^*\) of spaces with non-symmetric affine connections. Also, we investigate a special class of equitorsion almost geodesic mappings of type \(\pi _1^*\) and find some invariant geometric objects of these mappings.

Almost Geodesic Mappings of Spaces with Affine Connection

This paper is devoted to the further development of the theory of almost geodesic mappings of spaces with affine connection.

Geodesic mappings and differentiability of metrics, affine and projective connections

In this paper we study fundamental equations of geodesic mappings of manifolds with affine and projective connection onto (pseudo-) Riemannian manifolds with respect to the smoothness class of these geometric objects. We prove that the natural smoothness class of these problems is preserved.

Some relations in non-symmetric affine connection spaces with regard to a special almost geodesic mappings of the third type

We investigate two kinds of special almost geodesic mappings of the third type in this paper. We also find some relations for curvature tensors of almost geodesic mappings of the third type.

Conformal and geodesic mappings of generalized equidistant spaces

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.

Special equitorsion almost geodesic mappings of the third type of non-symmetric affine connection spaces

In this paper we investigate a special kind of almost geodesic mapping of the third type of spaces with non-symmetric affine connection. Also we find some relations for curvature tensors of associated affine connection spaces of almost geodesic mappings of the third type. Finally, we investigate equitorsion almost geodesic mapping having the property of reciprocity and find an invariant geometric object of this mapping.

On canonical almost geodesic mappings of the first type of affinely connected spaces

In this paper, we study special cases of canonical almost geodesic mappings of the first type of affinely connected spacesThe basic equations of mappings in question are reduced to a closed system of Cauchy type in covariant derivatives, and the number of essential parameters in the general solution of this system is estimated. We give an example of such mappings from a flat space onto another flat space. The mappings constructed send straight lines of the first space into parabolas in the second space. These almost geodesic mappings of the first type do not belong to the classes of mappings of the second and third types.

Geodesic mappings on compact riemannian manifolds with conditions on sectional curvature

We found new criteria for sectional curvatures on compact Riemannian manifolds for which geodesic mappings are affine, and, moreover, homothetic.

Geodesic mappings of (pseudo-) Riemannian manifolds preserve class of differentiability

Geodesic mappings of (pseudo-) Riemannian manifolds preserve class of differentiability

On geodesic mappings of equidistant generalized Riemannian spaces

In this paper geodesic mappings of equidistant generalized Riemannian spaces are discussed. It is proved that each equidistant generalized Riemannian space of basic type admits non-trivial geodesic mapping with preserved equidistant congruence. Especially, there exists non-trivial geodesic mapping of equidistant generalized Riemannian space onto equidistant Riemannian space. An example of geodesic mapping of an equidistant generalized Riemannian spaces is presented.