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Complex, contact, Riemannian, pseudo-Riemannian and Finsler geometry, relativity, gauge theory, global analysis.

0 votes
1 answer
94 views

A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for any …
oneyear's user avatar
  • 109
9 votes
2 answers
3k views

How to prove Liouville measure is invariant under geodesic flow?

Let $M$ be a complete n dimensional Riemannian manifold. $vol$ denotes the n dimensional Hausdorff measure. Let $$ SM=\{(x,v)|x\in M, v\in T_xM, \|v\|=1\} $$ be the unit tangent bundle of $M$. Then $S …
oneyear's user avatar
  • 109