Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces

Eugene D. Rodionov; Viktor V. Slavskii

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 271-282
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Rodionov, Eugene D., and Slavskii, Viktor V.. "Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 271-282. <http://eudml.org/doc/248998>.

@article{Rodionov2002,
abstract = {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.},
author = {Rodionov, Eugene D., Slavskii, Viktor V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {conformal deformations; Riemannian metrics; homogeneous Riemannian spaces; conformal deformations; Riemannian metrics; homogeneous Riemannian spaces},
language = {eng},
number = {2},
pages = {271-282},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces},
url = {http://eudml.org/doc/248998},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Rodionov, Eugene D.
AU - Slavskii, Viktor V.
TI - Conformal deformations of the Riemannian metrics and homogeneous Riemannian spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 271
EP - 282
AB - 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.
LA - eng
KW - conformal deformations; Riemannian metrics; homogeneous Riemannian spaces; conformal deformations; Riemannian metrics; homogeneous Riemannian spaces
UR - http://eudml.org/doc/248998
ER -

References

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  12. Rodionov E.D, Slavskii V.V., 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) MR1789653
  13. Rodionov E.D., Slavskii V.V., Locally conformally homogeneous Riemannian spaces, Journal of ASU 1 (19), (2001), 39-42. 
  14. Tricerri F., Locally homogeneous Riemannian manifolds, Rend. Semin. Mat. Torino 50 4 411-426 (1992). (1992) Zbl0793.53056MR1261452
  15. Tricerri F., Vanhecke L., Homogeneous structures on Riemannian manifolds, London Mathematical Society Lecture Note Series, 83; Cambridge etc.: Cambridge University Press, VI, 125 pp. Zbl0641.53047MR0712664
  16. Wallach N., Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. 96 277-295 (1972). (1972) Zbl0261.53033MR0307122
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