Answer by Stephen Miller for When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism?

2010-10-13T15:18:50Z

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).

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 $F$.

A good reference is Bredon: Compact Tranformation Groups

Answer by Stephen Miller for Nonconvex manhole covers

2010-07-22T10:40:50Z

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.

User stephen miller - MathOverflow

Answer by Stephen Miller for When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism?

Answer by Stephen Miller for Nonconvex manhole covers