Compute mean curvature of geodesic ball in $n$-sphere I want to compute mean curvature of boundary of geodesic ball in $S^n$.
I can calculate mean curvature of a ball in Euclidean space $(1/R)$.
However, I cannot calculate mean curvature in Riemannian manifold.
I want to regard sphere as an ambient space (not a submanifold of Euclidean space).
More precisely, $(S^n, g)$ denote a sphere, where $g$ is the warped product metric, i.e., $g= dr^2 + sin^2 r\, g_{S^n-1}$.  We assume that $r=0$ indicate the North Pole in sphere.
There is a $n$-dimensional geodesic ball centered at the North Pole with radius $R$.  Then could I compute mean curvature of the ball at the boundary of this ball?

If you define $u:\mathbb{S}^n\to\mathbb{R}$ to be the distance function from the North pole, we have the mean curvatura equal to
$$H=-div\left(\frac{\nabla u}{|\nabla u|}\right).$$
Thus, for a geodesic sphere of radius R, we obtain
$$H=\cot R$$
