In this paper we investigate one-dimensional sectional curvatures of Riemannian manifolds, conformal deformations of the Riemannian metrics and the structure of locally conformally homogeneous Riemannian manifolds. We prove that the nonnegativity of the one-dimensional sectional curvature of a homogeneous Riemannian space attracts nonnegativity of the Ricci curvature and we show that the inverse is incorrect with the help of the theorems O. Kowalski-S. Nikčevi'c [K-N], D. Alekseevsky-B. Kimelfeld [A-K]. The criterion for existence of the left-invariant Riemannian metrics of positive one-dimensional sectional curvature on Lie groups is presented. Classification of the conformally deformed homogeneous Riemannian metrics of positive sectional curvature on homogeneous spaces is obtained. The notion of locally conformally homogeneous Riemannian spaces is introduced. It is proved that each such space is either conformally flat or conformally equivalent to a locally homogeneous Riemannian space.
@article{119319,
author = {Eugene D. Rodionov and Viktor V. Slavskii},
title = {Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {271-282},
zbl = {1090.53039},
mrnumber = {1922127},
language = {en},
url = {http://dml.mathdoc.fr/item/119319}
}
Rodionov, Eugene D.; Slavskii, Viktor V. Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 271-282. http://gdmltest.u-ga.fr/item/119319/
The structure of homogeneous Riemannian spaces with zero Ricci curvature, Functional Anal. Appl. 9 (2), 5-11 (1975). (1975) | MR 0402650
Homogeneous Riemannian manifolds of positive Ricci curvature, Mat. Zametki 58 (3), 334-340 (1995). (1995) | MR 1368542
Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa 15 179-246 (1961). (1961) | MR 0133083 | Zbl 0101.14201
Einstein Manifolds, Springer-Verlag, Berlin, 1987. | MR 0867684 | Zbl 1147.53001
A comment on the integrability conditions of the conformal Killing equation, Gen. Relativity Gravitation 21 9 979-980 (1989). (1989) | MR 1008832 | Zbl 0675.53025
Conformal deformations and extremal paths in the space of Riemannian metrics, Math. Nachr. 72 137-140 (1976). (1976) | MR 0420703
Counter-example to the ``second Singer's theorem'', Ann. Global Anal. Geom. 8 2 211-214 (1990). (1990) | MR 1088512 | Zbl 0736.53047
Eigenvalues of locally homogeneous riemannian $3$-manifolds, Geom. Dedicata 62 65-72 (1996). (1996) | MR 1400981
Curvature of left invariant metric on Lie groups, Adv. Math. 21 293-329 (1976). (1976) | MR 0425012
Stability theorems in geometry and analysis, Mathematical Institute of the SB of RAS, Novosibirsk, 1996. | MR 1462616 | Zbl 0848.30013
Homogeneous Riemannian Z-manifolds, PhD. dissertation, Mathematical Institute of the SB of RAS, 1982. | MR 0610778 | Zbl 0472.53051
Conformal deformations of the Riemannian metrics with sections of zero curvature on a compact manifold, Rep. of Acad. Sci. 373 (3), (2000), 300-303. (2000) | MR 1789653
Locally conformally homogeneous Riemannian spaces, Journal of ASU 1 (19), (2001), 39-42.
Locally homogeneous Riemannian manifolds, Rend. Semin. Mat. Torino 50 4 411-426 (1992). (1992) | MR 1261452 | Zbl 0793.53056
Homogeneous structures on Riemannian manifolds, London Mathematical Society Lecture Note Series, 83; Cambridge etc.: Cambridge University Press, VI, 125 pp. | MR 0712664 | Zbl 0641.53047
Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 277-295 (1972). (1972) | MR 0307122 | Zbl 0261.53033
The Theory of Lie Derivatives and its Applications, North-Holland Publishing Co., Amsterdam; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 1957. | MR 0088769 | Zbl 0077.15802