Timeline for When is there a natural Riemannian metric whose measure preserves a self-diffeomorphism?
Current License: CC BY-SA 2.5
5 events
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| Jun 9, 2010 at 18:11 | comment | added | Deane Yang | @rpotrie: You're right about the regularity. I honestly know nothing about how to get an invariant measure with any given regularity (continuous, differentiable, or something else). I'm just observing that once you know what measure you want, the construction of a metric with the same regularity as the measure is completely straightforward. | |
| Jun 9, 2010 at 17:55 | comment | added | rpotrie | @Deane: You are right. The problem here is that one does not know if the density is even continuous, so, the metric can not be changed as easily. There is however a theorem by Oxtoby-Ulam (which can be seen as a continuous counterpart to moser´s theorem) which states that if a homeomorphism preserves a non atomic probability measure which is positive in open sets, then it is conjugated to a homeomorphism preserving Lebesgue measure. | |
| Jun 9, 2010 at 3:10 | comment | added | Deane Yang | Steve, if you want $f$ to be only measure-preserving and not an isometry, then once you have found an invariant measure, any metric whose volume form is equal to that measure meets your criteria. And it is easy to construct lots of metrics with a given volume form. | |
| Jun 8, 2010 at 21:14 | comment | added | Steve Huntsman | Thanks, it's nice to see this. But I'm really wondering about the metric more than the measure. | |
| Jun 8, 2010 at 20:48 | history | answered | rpotrie | CC BY-SA 2.5 |