Weak derivatives and Sobolev spaces on Riemannian manifolds

Question

I am decently experienced on Sobolev spaces on Euclidean spaces, but I just know basic ideas on Riemannian manifolds and want to understand something on Sobolev spaces on them.

Let $(M,g)$ be smooth Riemannian manifold of dimension $m$ and let $\phi:M \to \mathbb{R}^l$. For $x \in M$ and $v \in \mathbb{S}^{m}$ we write $\frac{d}{dt}\phi(\exp_x(tv))|_{t=0}=D\phi(x)v$ and $\frac{d^2}{dt^2}\phi(\exp_x(tv))|_{t=0}=Hess(\phi)(x)(v,v)$ and observe functionals $\int_M \int_{\mathbb{S}^m} |D\phi(x)v|^2 dv dx$ and $\int_M \int_{\mathbb{S}^m} |Hess(\phi)(x)(v,v)|^2 dv dx$. These functionals should be norms equivalent to $L^2(M)$-norms: $\int_M |D\phi(x)|^2 dx$ and $\int_M |Hess(\phi)(x)|^2 dx$, respectively.

Question. How do these $L^2(M)$-norms come into play in the definition of $\phi \in H^2(M,\mathbb{R}^l)$? More precise, can one define Sobolev norm as $\|\phi\|^2_{H^2(M)}=\|\phi\|^2_{L^2(M)} + \|D\phi\|^2_{L^2(M)}+\|Hess(\phi)\|^2_{L^2(M)}$ and $H^2(M)$ as closure of $C^\infty(M)$ with respect to that norm?

I have seen several different (yet equivalent) ways to define Sobolev spaces on Riemannian manifold, but it is not fully clear to me what kind of weak derivatives one observes.

UPDATE: A way to tackle this question is Proposition 1 in our paper.

Answer 1

Indeed, those norms are equivalent to the classical integrals of $|D\phi|^2$ and $|\mathrm{Hess} \phi|^2$. Then you can define $H^2(M, \mathbb{R}^l)$ as usual as the completion of the space of smoothly compactly supported fonctions. For nice manifolds (e.g. compact or asymptotically Euclidean ones) this is equivalent to the classical definition of functions whose weak first and second derivatives (same definition as on $\mathbb{R}^n$) belong to $L^2$. And, in practice (for me at least), there is no difference. For more recent results please check this paper: https://arxiv.org/abs/2011.14630

To answer your question in the comments, the Sobolev spaces are independent of the metric you choose as long as the metrics are uniformly equivalent (i.e. there exists a constant $C > 1$ such that $C^{-1} g \leq h \leq C g$), leaving aside regularity questions. The main difficulty when learning Sobolev spaces on manifolds is that there is no definite reference to read and this property among others is something which you have to figure out yourself. The proof is by no means complicated, it is based on the fact that if you have two connections (say Levi-Civita connections), e.g. $\nabla$ and $D$ (associated to $g$ and $h$ respectively), then they differ by a first order term:

• For functions the difference is zero as we have $\nabla f = Df = df$ (the usual differential).
• For 1-forms (e.g. $df$) the difference can be expressed in terms of the "Christoffel symbols": $$ \nabla_i X_j = D_i X_j - \Gamma_{ij}^k X_k $$ where $$ \Gamma_{ij}^k = \frac{1}{2} g^{kl} \left(D_i h_{lj} + D_j h_{il} - D_l h_{ij}\right) $$ (this corresponds to the usual Christoffel symbols if $h$ is Euclidean). I can explain better later if you need me to provide the details.