Is the space of Riemannian metrics, over a compact manifold, complete when endowed with the $C^k$topology of metrics?. Is there a good reference for this?
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No it is not complete. It is an open convex cone of the Banach space (Frechet space if $k=\infty$) $\Gamma_{C^k} S^2T^*M$ of $C^k$sections of the vector bundle $S^2T^*M$. 0 is always in the closure of this cone, and many more things. The norm on this Banach space depends on many choices (charts, metric, etc.), but all these norms are equivalent. You can also put several natural Riemannian metrics on the space of all Riemannian metrics, but none of them is geodesically complete. Natural means: invariant under the diffeomorphism group. See: Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. 20 pages. To appear in: Journal of Differential Geometry.(pdf) Check also the references there. 

