
Journal of Lie Theory 18 (2008), No. 1, 243251 Copyright Heldermann Verlag 2008 Some Basic Results Concerning Ginvariant Riemannian Metrics Marja Kankaanrinta Dept. of Mathematics, PO Box 400137, University of Virginia, Charlottesville, VA 229044137, U.S.A. mk5aq@virginia.edu [Abstractpdf] we study complete $G$invariant Riemannian metrics. Let $G$ be a Lie group and let $M$ be a proper smooth $G$manifold. Let $\alpha$ be a smooth $G$invariant Riemannian metric of $M$, and let $\tilde{K}$ be any $G$compact subset of $M$. We show that $M$ admits a complete smooth $G$invariant Riemannian metric $\beta$ such that $\beta\vert \tilde{K}=\alpha\vert \tilde{K}$. We also prove the existence of complete real analytic $G$invariant Riemannian metrics for proper real analytic $G$manifolds. Moreover, we show that for any given smooth (real analytic) $G$invariant Riemannian metric there exists a complete smooth (real analytic) $G$invariant Riemannian metric conformal to the original Riemannian metric. To prove the real analytic results we need the assumption that $G$ can be embeddded as a closed subgroup of a Lie group which has only finitely many connected components. Keywords: Lie groups, Riemannian metric, real analytic. MSC: 57S20 [ Fulltextpdf (164 KB)] for subscribers only. 