In a d-dimentional Riemannian Manifold $(M,g)$, we have a ball with radius $r$, where the radius is defined as the geodesic distance from center to the boundary of the ball. How to compute the volume of it using the volume form? Intuitively, it should be $kr^d$, where $k$ should be some curvature depending constant. How to show this in general?
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1$\begingroup$ If you know nothing about the curvature, your formula cannot be true. $\endgroup$– user99914Mar 25, 2017 at 7:26
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As John pointed out in the comments, your intuition is incorrect. Small balls have approximate volume $kr^d$ where $k$ depends only on $d$ (not on curvature!), but in general there is no easy formula for the exact volume - you need to integrate the volume form over the ball. If you know the scalar curvature then you can get a better approximation - see e.g. here.
answered Mar 25, 2017 at 14:36