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!


1 Answer 1


If you're defining Sobolev spaces using weak derivatives, you will need a corresponding notion for Riemannian manifolds. Luckily, this is not too difficult to do. While the process will depend on your precise choice of definitions, the general rule is to work locally with charts.

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.

  • $\begingroup$ 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? $\endgroup$
    – user500357
    May 9 at 22:00
  • $\begingroup$ @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. $\endgroup$
    – Kajelad
    May 10 at 17:37

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