Abstract: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 t… Show more
“…In this paper, we study canonical almost geodesic mappings of the first type of spaces with an affine connection onto generalized m-Ricci-symmetric spaces. These spaces are the essential generalization of the spaces considered in [24]. Ricci tensor is well known from the Einstein equation that describes gravitational fields.…”
Section: Discussionmentioning
confidence: 99%
“…Obviously, in the space A n the Equations ( 3), ( 20)-( 22), ( 32) and ( 33) form a closed system of PDEs of Cauchy type with respect to the functions P h ij (x), a ij (x), R h ijk (x), R h ijkm (x), a ijk (x), a ijkm (x). Also the functions must satisfy the algebraic conditions (24).…”
Section: Canonical Almost Geodesic Mappings Of Type π 1 Of Spaces Witmentioning
confidence: 99%
“…Let us differentiate covariantly (m − 3) times the Equation ( 27) with respect to the connection of the space A n . Then express in the left-hand side the covariant derivatives with respect to the connection of the space A n in terms of the covariant derivatives with respect to the connection of the space A n , using the formula [24]…”
Section: Canonical Almost Geodesic Mappings Of Type π 1 Of Spaces Witmentioning
confidence: 99%
“…Sobchuk [19,20], N.Y. Yablonskaya [21,22], V.E. Berezovski, J. Mikeš [14,15,[23][24][25][26][27][28][29][30][31][32][33][34][35], O. Belova, J. Mikeš, K. Strambach [36,37], M.S. Stankovič, Lj.S.…”
Section: Introductionmentioning
confidence: 99%
“…Let us note that in works [7,8,24,[46][47][48][49][50] were studied geodesic and holomorphically projective mappings of above mentioned spaces. For the other mappings the main equations are obtained as closed systems of PDEs of Cauchy type, see [5,6,[51][52][53][54].…”
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš.
“…In this paper, we study canonical almost geodesic mappings of the first type of spaces with an affine connection onto generalized m-Ricci-symmetric spaces. These spaces are the essential generalization of the spaces considered in [24]. Ricci tensor is well known from the Einstein equation that describes gravitational fields.…”
Section: Discussionmentioning
confidence: 99%
“…Obviously, in the space A n the Equations ( 3), ( 20)-( 22), ( 32) and ( 33) form a closed system of PDEs of Cauchy type with respect to the functions P h ij (x), a ij (x), R h ijk (x), R h ijkm (x), a ijk (x), a ijkm (x). Also the functions must satisfy the algebraic conditions (24).…”
Section: Canonical Almost Geodesic Mappings Of Type π 1 Of Spaces Witmentioning
confidence: 99%
“…Let us differentiate covariantly (m − 3) times the Equation ( 27) with respect to the connection of the space A n . Then express in the left-hand side the covariant derivatives with respect to the connection of the space A n in terms of the covariant derivatives with respect to the connection of the space A n , using the formula [24]…”
Section: Canonical Almost Geodesic Mappings Of Type π 1 Of Spaces Witmentioning
confidence: 99%
“…Sobchuk [19,20], N.Y. Yablonskaya [21,22], V.E. Berezovski, J. Mikeš [14,15,[23][24][25][26][27][28][29][30][31][32][33][34][35], O. Belova, J. Mikeš, K. Strambach [36,37], M.S. Stankovič, Lj.S.…”
Section: Introductionmentioning
confidence: 99%
“…Let us note that in works [7,8,24,[46][47][48][49][50] were studied geodesic and holomorphically projective mappings of above mentioned spaces. For the other mappings the main equations are obtained as closed systems of PDEs of Cauchy type, see [5,6,[51][52][53][54].…”
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš.
We prove a number of nonexistence theorems for metrics projectively and
holomorphically projectively equivalent to parabolic metrics and metrics of
finite volume of complete Riemannian and K?hler manifolds, respectively.
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