User stephen miller - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:52:25Z http://mathoverflow.net/feeds/user/7830 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27511/when-is-there-a-natural-riemannian-metric-whose-measure-preserves-a-self-diffeomo/42028#42028 Answer by Stephen Miller for When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism? Stephen Miller 2010-10-13T15:18:50Z 2010-10-13T15:18:50Z <p>A similar question occurs in the theory of topological groups. Given a measure $\mu$ on a $G$-space X, can one use $\mu$ to construct an $equivariant$ measure? The answer is yes if $G$ is compact: you define $\bar{\mu}(U)$ by integrating $\mu(gU)$ over $g\in G$, with respect to the (essentially unique) equivariant measure on $G$. If $G$ is finite, that's just the average over $g\in G$ (you usually assume the measure on $G$ integrates to one).</p> <p>In your case, you could consider the subgroup of $Diff(M)$ generated by your diffeomorphism $f$. If this is compact, or more likely if $f$ lies in any compact $F\subset Diff(M)$, you should be able to define an integral of your metric $g$ over $F$.</p> <p>A good reference is Bredon: Compact Tranformation Groups</p> http://mathoverflow.net/questions/31810/nonconvex-manhole-covers/32934#32934 Answer by Stephen Miller for Nonconvex manhole covers Stephen Miller 2010-07-22T10:40:50Z 2010-07-22T10:40:50Z <p>In the non-simply connected case, an annulus gives an easy example. To get a simply connected example, just cut a thin strip out of the annulus along a radius.</p>