Let $(M,g)$ be a compact oriented Riemannian manifold and $E\to M$ be a vector bundle with metric $h$ and a connection $\nabla$. Then one define the sobolev space $W^{k,p}(E)$ as the sets of $L^p$ section $u$ whose weakly covariant differential $\nabla^ju$, $j\le k$, belongs to $L^p$. More precisely, the weakly differential $\nabla u$ is defined by $\langle \nabla u,\varphi\rangle=\langle u,\nabla^*\varphi\rangle$ for each $φ\in C^\infty_c(T^*M\otimes E)$, where $\nabla^*$ is the formal adjoint and the pair is given by integration.
Partial differential operator $\nabla$ is given by a combination of partial derivatives $\sum A_{ij}(x)\frac{\partial}{\partial x^i}$ locally. It is clear the combination is far different with each partial derivative. Can you show me some tips?
Finally, the question is how to prove the space $W^{k,p}$ is independent with the metrics and connection. But please note that the space is not defined as the completion of smooth section with $L^p$ differential and the diffcult ocurs when one try to approximate a section by smooth one cause the notion of weakly differential. Any reference is welcome. Thanks a lot!
Best Answer
I think you do not need to use weak derivatives; it is enough to use the standard Soblev $k$-norm on the Fourier side. But I do not think you can cast the metric aside, though you can use the fact that the set of metrics on a Riemannian manifold is path connected. Then you can pushing around the inequalities by partition of unity and a continuity argument using the geodesic coorindate. Since any two connection differ by an $End(TM)$ value one form, you should be able to show that this is not dependent on the choice of connections.
It is not clear to me what is the standard reference on this topic. I think Peterson's book might be helpful, but I never read it so cannot comment on it.