2

I want to get a feeling for how much flexibility we have when putting a Riemannian metric on a given smooth manifold $M$.

Is it always possible to find two non-isometric metrics on $M$? If the answer is positive, is there some qualitative\quantitative estimate on how many different (non-isometric) metrics exists on $M$?

I understand the set of all Rimeannian manifolds might be an infinite dimensional manifold but I think this notion distinguishes between two isometric metrics, while I am asking with "how many" metrics we are left with after we identify those which are isometric.

Community
  • 1
Asaf Shachar
  • 25,967
  • I would guess there are always many non-isometric metrics (where by "many" I probably mean uncountably many). There are probably easier ways to see this, but certainly (at least if $M$ is compact) there are results on prescribed scalar curvature that imply the class of possible scalar curvature functions is large. For example, if $M$ is compact, it seems that one could take a constant scalar curvature metric (possible by the solution to the Yamabe problem), and perturb it in a small open set to obtain many non-isometric metrics. – Phillip Andreae Apr 22 '15 at 17:31

0 Answers0