I was just remembering my old course on Riemannian geometry and a question went through my mind regarding the tangent space.
Giving you manifold a Riemannian metric makes the tangent space at each point into a pre-Hilbert space. Is this space Hilbert space, i.e do we have that each tangent space is complete ?
And if not, can we build a Riemannian manifold and embedding such that the tangent space at each point of the new manifold is simply the completion of the tangent space at the preimage and such that the pushforward map at each point is the "canonical map" that sends a pre-Hilbert space into its completed space (i.e. it's an isometric map such that the image of the map is dense in the codomain if I remember correctly how the completion of a metric space works). Also is this unique up to a unique isomorphism (I don't remember how are called isomorphisms in the category of Riemannian manifolds).
To be more precise, if $M$ is our original manifold, does there exists an embedding $\iota:M\rightarrow \tilde{M}$ such that $(\iota^*)_p:T_pM\rightarrow T_{\iota(p)}\tilde{M}$ is an isometric map and $(\iota^*)_p(T_pM)$ is dense in $T_{\iota(p)}\tilde{M}$. Also, if the pair ($\iota,\tilde{M}$) exists, is it up to a unique isomorphism ?
I hope my question makes sens. All of those notions are kind of blurry nowadays as I don't really do math (or geometry let's say) anymore.