Volumes and metric

Question

I have a pretty general question regarding volumes of manifolds and metrics. I was wondering if knowledge of the different volumes and how they relate to each other can tell you anything about the metrics that calculate the volume. For example, what can we say about metrics that give the same volume? Are they isometric? Or what can we say about metrics that give different volumes? Do lengths of curves from the metric that gives greater volume exceed the lengths of curves from the metric that gives lesser volume? Thank you.

Answer 1

I was wondering if knowledge of the different volumes and how they relate to each other can tell you anything about the metrics that calculate the volume. 

Not very much - the space of volumes is much smaller than the space of metrics: the space of volume forms is something like (a double cover of) the space of functions on the manifold, whereas the space of metrics has about n(n+1)/2 independent functions (space of symmetric two-tensors subject to constraints of being a metric).

For example, what can we say about metrics that give the same volume? Are they isometric? 

Definitely not. Take any two metrics on the 2-sphere (say the usual round one with curvature normalized to some value, and another one, say with a region of negative curvature). By scaling one by some constant value they can be arranged to have the same volume, but are clearly very different (certainly not isometric).

Or what can we say about metrics that give different volumes? Do lengths of curves from the metric that gives greater volume exceed the lengths of curves from the metric that gives lesser volume? 

Again, not necessarily. Take a two sphere and pinch it down along the equator, then blow the metric up by scaling it by a (large) constant. In this way you can construct metrics with very short (even geodesic) curves but very large volumes (say compared to the round metric of some fixed curvature).