A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.
In this paper we study the theory of F-planar mappings of spaces with affine connection. We obtained condition, which preserved the curvature tensor. We also studied canonical F-planar mappings of space with affine connection onto symmetric spaces. In this case, the main equations have the partial differential Cauchy type form in covariant derivatives. We got the set of substantial real parameters on which depends the general solution of that PDE's system.
In the paper, we consider geodesic mappings of spaces with an affine connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an affine connections onto 2-, 3-, and m- (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an affine connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m- (Ricci-) symmetric spaces (m≥1) are geodesically mapped onto many spaces with an affine connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions.
In this paper we study fundamental equations of holomorphically projective mappings of parabolic Kähler spaces (which are generalized classical, pseudo-and hyperbolic Kähler spaces) with respect to the smoothness class of metrics. We show that holomorphically projective mappings preserve the smoothness class of metrics.
We consider conformal and concircular mappings of Eisenhart's generalized Riemannian spaces. We prove conformal and concircular invariance of some tensors in Eisenhart's generalized Riemannian spaces. We give new generalizations of symmetric spaces via Eisenhart's generalized Riemannian spaces. Finally, we describe some properties of covariant derivatives of tensors analogous to Yano's tensor of concircular curvature in Eisenhart symmetric spaces of various kinds.
We investigate equitorsion holomorphically projective mappings of generalized m-parabolic Kähler manifolds and provide some necessary and sufficient conditions for the existence of these mappings in form of linear PDE-systems. Also, we find an invariant geometric object with respect to a holomorphically projective mapping of generalized m-parabolic Kähler manifolds which is analogous to the Thomas projective parameter.
Definition 1.1. [18]A generalized Riemannian manifold (M, ) of even dimension n (n > 2) is said to be a generalized m-parabolic Kähler manifold if there exists a tensor field F on M of type (1, 1) such that rank(F) = m ≤ n 2 and the following conditions hold
In this paper, we study n-dimensional recurrent equiaffine projective Euclidean manifolds, i.e. manifolds with absolute recurrent curvature tensor, which admit geodesic mappings onto Euclidean space, and they are equiaffine (where was obtained the symmetric Ricci tensor). We obtained main conditions of recurrent projective Euclidean spaces and constructed their examples.
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