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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!

MiGang
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1 Answers1

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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.

Bombyx mori
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  • Thanks for your help. I want to find a down-to-earth explanation why the existence of weakly covariant differential can imply that the section is weakly first order differentiable in a section. – MiGang Aug 23 '14 at 13:23
  • Well, this should be left to you. – Bombyx mori Aug 23 '14 at 15:55
  • :The weakly covariant differential is defined via formal adjoint. Locally the partial differential operator $\nabla$ is goven by a combination of partial derivatives $\sum A_{ij}(x)\frac{\partial}{\partial x^i}$. Now the weakly differential $\nabla u$ is defined by $\langle \nabla u,\varphi\rangle=\langle u,\nabla^\varphi\rangle$ for each $\varphi\in C^\infty_c$, where $\nabla^$ is the foemal adjoint and the pair is given by integration. It is clear the combination is far different with each partial derivative. Can you show me some tips? – MiGang Aug 24 '14 at 00:00
  • Locally $\nabla$ is $\bf{not}$ given by the combination of partial derivatives. It is of the form $d+\omega$. I suggest you review the basic definition first. – Bombyx mori Aug 24 '14 at 00:06
  • I missed the $0$-order terms. Excuse me, doesnot one confront with the same diffcult provided the space is not defined by completion of smooth section? – MiGang Aug 24 '14 at 00:24
  • There is no zero order term for $\nabla$. I suggest you check the textbook. – Bombyx mori Aug 24 '14 at 00:50