# Higher order derivatives for manifold valued Sobolev functions

Equipped with only very limited differential geometry knowledge, I am currently trying to understand Sobolev spaces of manifold valued functions $$f:\Omega \subset \mathbb{R}^d \to M$$, where $$M$$ is a Riemannian manifold. There seem to be multiple ways to define these spaces, a very common one is based on the Nash embedding theorem. With embedding $$\iota$$ one sets: $$W_{\iota}^{k,p}(\Omega,M)=\{v\in W^{k,p}(\Omega,\mathbb{R}^N): v(x)\in \iota(M)\}.$$ In some articles I read, people then define higher order derivatives for a multiindex $$\beta\in\{1,\dots ,d\}^k$$ by $$\nabla^{\beta}u:=\nabla_{du_{\beta_k}}\dots \nabla_{du_{\beta_2}}du_{\beta_1},$$ where $$\nabla_{du}$$ should be covariant derviatives along $$u$$ for $$u\in C(\Omega,M)\cap W^{k,p}(\Omega,M)$$.

If $$u$$ would be a $$C^k(\Omega,M)$$ function (defined in the usual way via local charts), I would understand this definition: The first derivative would be the usual differential of a manifold valued function and afterwards we have covariant derivatives along u. As the local coordinate representation of $$u$$ is $$C^k$$, everything is well defined. However, with $$u\in W^{k,p}\cap C(\Omega,M)$$ (defined via the embedding above) I do not understand this definition at all: How do you even explain $$du_{\beta_1}$$, how do you explain a covariant derivative along such an $$u$$ and how can you make sure that $$du_{\beta_1}$$ is sufficiently regular to be differentiated covariantly again? As the authors in these articles do not even comment on such questions, I probably overlook some very easy argument, so any help would be highly appreciated. Thanks in advance!

More precisely, given a continuous function $$f:M\to N$$ between smooth manifolds, we say that $$f$$ is weakly $$k$$-times differentiable if the local representative of $$f$$ w.r.t. any pair of coordinate charts on $$M$$ and $$N$$ is weakly $$k$$-differentiable as a map between open sets in $$\mathbb{R}^n$$. Different choices of charts will agree on their common domain.
You can build a global theory of Sobolev spaces of maps between Riemannian manifolds, and things proceed more or less exactly as in $$\mathbb{R}^n$$, but this requires a bit more work.
• Ok, I think I understand the idea to define k-times weakly differentiable by postulating that the coordinate representation is k-times weakly differentiable. However, why is it guaranteed that a function in $W_{\iota,M}^{k,p}(\Omega)\cap C(\Omega,M)$ as defined above with the help of the embedding has a coordinate representation which is k-times weakly differentiable? May 9 at 22:00
• @user500357 In the usual setting of open sets in $\mathbb{R}^n$, if $u$ is weakly $k$-differentiable and $f$ is an embedding, then $f\circ u$ is also weakly $k$-differentiable. The coordinate representations can be written as compositions of this form. May 10 at 17:37