# Definition of Hölder Spaces of maps between manifolds (also Sobolev Spaces of these maps)

Let $M$ be a Riemannian manifold, $N$ a manifold, $k\in \mathbb{N}$, $\alpha\in(0,1)$

Question: How exactly is the Hölder space $C^{k,\alpha}(M,N)$ and the sobolev space $W^k_p(M,N)$ defined? Are there (good) references where these spaces are introduced and sobolev embedding theorems of these spaces can be found?

(I know how $C^{k,\alpha}(M,\mathbb{R})$, $W^k_p(M,\mathbb{R})$ and $C^{k,\alpha}(E)$, $W^k_P(E)$ for sections of a vector bundle $E\rightarrow M$ are defined if that helps.)

Also I would appreciate any good references on the topic of sobolev and hölder spaces of maps between manifolds and sections of vector bundles.

• For Sobolev spaces, the only way I know how to make sense of this would be to embed $N$ in some ambient Euclidean space $\mathbb{R}^N$, and then define $W^{k,p}(M,N):=\{u\in W^{k,p}(M,\mathbb{R}^N):u(x)\in N \text{ a.e.}\}$. – Matt Jun 11 '15 at 18:06
The Hölder maps are very straightforward to define in the usual way if you have a metric on both the domain and range. If $N$ is just a differentiable manifold, I guess the local version of Hölder continuity is invariant under local diffeomorphisms. So that would still be meaningful, although not the global version.