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Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold
$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
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Embedding Riemmanian Manifold Linearly
Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
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