Let M be a simple connected Riemannian Manifold wih $K_M < 0$. Prove that the volume of any geodesic ball of M is strictly greater than $\frac{Vol(S^{n-1})r^n}{n}$, where $n = dim(M)$ and $r$ is the radius of the ball.

What I know: Since the manifold is a Hadamard Manifold the exponencial map is a diffeomorfism. So using $exp_p: T_pM \to M$ as a coordinate chart we have a global chart for any geodesic ball.

I know how to caculate the volume of a ball in $T_pM$, Ie, in a Riemannian manifold with $K=0$, and this one is exactly $\frac{Vol(S^{n-1})r^n}{n}$, but I'm having difficult in comparing this volume with the volume in M.

Thank you!


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