$M$ is n-dimensional Riemannian manifold, and has constant sectional curvature $K_0$. When $r>0$ is small enough, denote $$ B(p,r) = \exp_p(B(r)) $$ where $B(r)\subset T_pM$ is a ball of radius $r$ and centered on the origin. I know the volume of $B(p,r)$ has form $$ Vol(B(p,r)) = C(K_0) r^n $$ where $C(K_0)$ is constant depending on $K_0$. This morning, I try to prove it. But fail. And seemly, in do Carmo's Riemannian Geometry, there is not an example about this from the point of volume form. Therefore, I want to know how to calculate it or where I can find the calculation. Thanks.
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1The formula you've written is incorrect; The volume is only proportional to $r^n$ when $K_0=0$. As for computing the correct form, are you familiar with the Jacobi equation? – Kajelad May 17 '22 at 01:54
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@Kajelad I am familiar with Jacobi equation (or Jacobi field). – Enhao Lan May 17 '22 at 02:49