I am browsing through a paper and I am confused by a notation the meaning of which I do not understand.
Let $S_{p,\rho}$ be the geodesic sphere of center $p$ and radius $\rho$ in a Riemannian manifold $(M,g)$ of $\dim \ M = m+1$ (with $\rho$ less than the injectivity radius at $p$). Let $S^m$ be the unit sphere in $T_p M$; note that there exist the global parametrization $S^m \to S_{p,\rho}$ given by $\Theta \mapsto \exp _p (\rho \Theta)$.
If $H(S_{p,\rho})$ is the total mean curvature of $S_{p,\rho}$ (which I suspect to be the integral over $S_{p,\rho}$ of the mean curvature at every point, but I am not sure), then the first formula on page 2 of the paper claims that
$$H(S_{p,\rho}) = \frac m \rho - \frac 1 3 Ric_p (\Theta, \Theta) \rho + O(\rho ^2) .$$
What I do not understand here is why the coefficient of $\rho$ depends on a point $\Theta \in S^m$ (which one?) when the left-hand side should be independent of it. Shouldn't there maybe be some integration on $S^m$ to take care of that $\Theta$? What am I not understanding?