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I'm studying Jurgen Jost's Riemannian geometry and geometric analysis, in the appendix there is a version of Poincare's inequality on compact Riemannian manifold: if $M$ is a compact oriented Riemannian manifold, if $f\in H^1(M)$ (the Sobolev space) satisfies $\int_M f=0$, then $$||f||_{L^2(M)}\leq C||Df||_{L^2(M)}.$$ Here $C$ is some positive constant depending on $M$ only.

I googled but could not find a reference for this result. It doesn't seem to follow directly from the version on open sets of $\mathbb{R}^n$ because when you take partition of unity $\phi_\alpha$, the derivative on the right hand side messes things up. Can someone give me a reference or point out how to proceed? Thank you!

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The usual references are the books from Emmanuel Hebey: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities and Sobolev Spaces on Riemannian Manifolds.

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