I have the following basic question about Sobolev-spaces which take their values in a
Riemannian manifold $(M,g)$, i.e.
functions $u:\Omega \to M$, $\Omega \subset \mathbb{R}^n$ bounded, such that for every chart $\varphi : M \to \mathbb{R}^d$, the composition $\varphi\circ u$ is in the usual $H^1$ space.
Question: 1. Is this notion well-defined (this is only easy to show for higher order smoothnes or $n =1$)?
2. Does the manifold-valued Sobolev space admit the structure of a Hilbert manifold?
3. Do there exist good references for manifold-valued Sobolev spaces (and their relations with Hilbert manifolds)?
4. Is there a better way to define manifold-valued Sobolev spaces?
Many thanks for your answers!
Best Answer
To my knowledge the usual approach to define manifold valued spaces is to embed the target manifold $M$ in some $R^N$, then consider the space of all functions $u:\Omega\to R^N$ which are in $H^1(\Omega,R^N)$ and in addition take their values in $M$: $u(x)\in M$ for a.e. $x\in \Omega$. For a basic example of this approach see the paper by Brezis and Mironescu (On some questions of topology for $S^1$ valued fractional Sobolev spaces) where functions with values in $S^1$ are considered. You will agree that in that case a very natural definition is to put a priori the constraint $|u|=1$ on the functions. Of course, you end up with considering bounded functions only, but this solves most of the problems you are facing with your definition.