The existence of homogeneous geodesics in homogeneous pseudo-Riemannian and affine manifolds

JOURNAL OF GEOMETRY AND PHYSICS

卷:60

Article

Dusek, Zdenek

关键词: Homogeneous manifold;    Pseudo-Riemannian manifold;    Affine connection;    Killing vector field;    Homogeneous geodesic;   

DOI  :  10.1016/j.geomphys.2009.12.015

来源: Elsevier

【 摘 要 】

It is well known that, in any homogeneous Riemannian manifold, there is at least one homogeneous geodesic through each point. For the pseudo-Riemannian case, even if we assume reductivity, this existence problem is still open. The standard way to deal with homogeneous geodesics in the pseudo-Riemannian case is to use the so-called Geodesic Lemma, which is a formula involving the inner product. We shall use a different approach: namely, we imbed the class of all homogeneous pseudo-Riemannian manifolds into the broader class of all homogeneous affine manifolds (possibly with torsion) and we apply a new, purely affine method to the existence problem. In dimension 2, it was solved positively in a previous article by three authors. Our main result says that any homogeneous affine manifold admits at least one homogeneous geodesic through each point. As an immediate corollary, we prove the same result for the subclass of all homogeneous pseudo-Riemannian manifolds. (C) 2010 Elsevier B.V. All rights reserved.

【 授权许可】

Free