A note on geodesic and almost geodesic mappings of homogeneous Riemannian manifolds

• Autorzy: Formella S.
• Języki publikacji: EN

Abstrakty

EN

Let M be a differentiable manifold and denote by nabla and nabla~ two linear connections on M. Nabla and nabla~ are said to be geodesically equivalent if and only if they have the same geodesics. A Riemannian manifold (M, g) is a naturally reductive homogeneous manifold if and only if nabla and nabla~ = nabla - T are geodesically equivalent, where T is a homogeneous structure on (M, g) ([7]). In the present paper we prove that if it is possible to map geodesically a homogeneous Riemannian manifold (M, g) onto (M, nabla~), then the map is affine. If a naturally reductive manifold (M, g) admits a nontrivial geodesic mapping onto a Riemannian manifold (formula) then both manifolds are of constant cutvature. We also give some results for almost geodesic mappings (M, g) arr (M, nabla~).

Słowa kluczowe

EN

homogeneous Riemannian manifold; geodesic; almost geodesic; geodesic mapping; almost geodesic mapping

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2005
• Tom: Vol. 25, no. 2
• Strony: 181--187
• Opis fizyczny: Bibliogr. 8 poz.

Twórcy

autor

Formella S.

• Technical University, Institute of Mathematics, al. Piastów 48/49, 70-310 Szczecin, Poland,

Bibliografia

• [1] Formella S.: On geodesic mappings in some Riemannian and pseudo-Riemannian manifolds. Tensor (N.S.), 46 (1987), 311-115.
• [2] Formella S.: On some class of nearly conformally symmetric manifolds. Coll. Math. Vol. LXVIII 1995. Fasc. 1 149-164.
• [3] Mikeś J.: Geodesic mappings of affine-connected and Riemannian spaces. New York, J. Math. Sci. 1996, 311-333.
• [4] Mikeś J.: Holomorphically projective mappings and their generalizations. New York, J. of Math. Sci. 89 (1998) 3, 1334-1353.
• [5] Sinyukov N.S.: Geodesic mappings of Riemannian Spaces. Moscow, Nauka, 1979 (in Russian).
• [6] Sobchuk V. S.: On geodesic mappings of generalized Ricci symmetric Riemannian manifolds. Univ. Chernovtsy, 1981.
• [7] Tricerri F., Vanhecke L., Homogeneous Structure on Riemannian Manifolds. Lon­don Math. Soc. Lecture Note Series, vol. 83, Cambridge Univ. Press 1983.
• [8] Yablonskaya N.V.: On certain classes of almost geodesic mappings of general affine-connected spaces. Ukr. Geom. Sb. (1984) 27, 120-124