Journal of Lie Theory

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.