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!
