# Space of $G$-invariant Riemannian metrics contractible?

A well-known result in (psuedo)Riemannian geometry is that the moduli space of (pseudo)Riemannian metrics on a smooth manifold is contractible. In the case when you have a smooth action of a group $G$ on a manifold $M$, is it obvious that the analogous result holds for $G$-invariant Riemannian metrics (say $G$ is a compact Lie group) on $M$? Seems like it should hold... on that note, I suppose a good, additional question would be: are there any any good references on $G$-invariant-type results?

• By $G$-invariant do you mean left-invariant? Or bi-invariant? – Travis Jun 4 '15 at 16:49
• @Travis Let's go with left-invariant for now. – ChickenSocks Jun 4 '15 at 17:01
• A good source for some results about left-invariant metrics is: Milnor, J., Curvatures of Left-Invariant Metrics on Lie Groups, Advances in Mathematics, 21 (3), September 1976, pp. 293–329. sciencedirect.com/science/article/pii/S0001870876800023 – Travis Jun 4 '15 at 17:23