A note on the Osserman conjecture and isotropic covariant derivative of curvature期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

按检索

A note on the Osserman conjecture and isotropic covariant derivative of curvature

Authors:	Novica Blazic Neda Bokan Zoran Rakic

Institution:	Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia ; Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia ; Faculty of Mathematics, University of Belgrade, Studentski trg 16, p.p. 550, 11000 Belgrade, Yugoslavia

Abstract:	Let be a Riemannian manifold with the Jacobi operator, which has constant eigenvalues, independent on the unit vector $X\in T_{p}M$ and the point $p\in M$ . Osserman conjectured that these manifolds are flat or rank-one locally symmetric spaces ( $\nabla R =0$ ). It is known that for a general pseudo-Riemannian manifold, the Osserman-type conjecture is not true and 4-dimensional Kleinian Jordan-Osserman manifolds are curvature homogeneous. We show that the length of the first covariant derivative of the curvature tensor is isotropic, i.e. $\Vert \nabla R\Vert =0$ . For known examples of 4-dimensional Osserman manifolds of signature we check also that $\Vert \nabla R\Vert =0$ . By the presentation of a class of examples we show that curvature homogeneity and $\Vert \nabla R\Vert =0$ do not imply local homogeneity; in contrast to the situation in the Riemannian geometry, where it is unknown if the Osserman condition implies local homogeneity.

Keywords:	Pseudo-Riemannian manifold curvature tensor Jacobi operator Kleinian Osserman spacelike (timelike) manifold Osserman conjecture isotropicity

	点击此处可从《Proceedings of the American Mathematical Society》浏览原始摘要信息
	点击此处可从《Proceedings of the American Mathematical Society》下载免费的PDF全文

设为首页 | 免责声明 | 关于勤云 | 加入收藏