I have been studying the book Some Nonlinear Problems In Riemannian Geometry - Thierry Aubin. On page $46$ he begins the proof of the Sobolev imbedding theorem to manifolds. The proof is divided in two steps, when $d=dist(p,q)\geq\delta$ and when $d<\delta$. In the second step ($d<\delta$), he considers a ball $\tilde{B}=B(O,d/2)$ such that $p,q\in\tilde{B}$ and defines the function $h(x)=f(\mbox{exp}_{O}(x))$ in $\tilde{B}$. After some manipulations, he uses the theorem 1.53 ( to obtain the inequality $$ \left(\int_{\tilde{B}}|\nabla_{E}h(x)|^{q}dE\right)^{1/q}\leq\frac{\sinh(a\delta)}{a\delta}\left(\frac{\pi}{2}\right)^{(n-1)q}\|\nabla f\|_{q} $$ where $\nabla_{E}$ is the Euclidean gradient. (Suppose that the assumptions of the theorem are satisfied and try to apply the theorem to obtain the inequality.)
Theorem 1.53: Let $M$ be a Riemannian manifold whose sectional curvature $K$ satisfies the bounds $-a^2\leq K\leq b^2$, the Ricci curvature being greater than $a'=(n - 1)\alpha^2$. Let $S_{P}(r_o)$ be a ball of $M$ with center $P$ and radius $r_o< \delta p$ the injectivity radius at P. Consider $(S_{P}(r_o), \exp_{P}^{-1})$, a normal geodesic coordinate system. Denote the coordinates of a point $Q = (r,\theta)\in[0, r_o]\times S_{n-1}(1)$, locally by $\theta = \{\theta^{i}\}$, ($i=1, 2,. . ., n - 1$). The metric tensor $g$ can be expressed by $$ ds^2=(dr)^2+r^2g_{\theta^{i}\theta^{j}}(r,\theta)d\theta^{i}d\theta^{j} $$ For convenience let $g_{\theta\theta}$ be one of the components $g_{\theta^{i}\theta^{i}}$ and $|g|=\mbox{det}((g_{\theta^{i}\theta^{j}}))$. Then $g_{\theta\theta}$ and $|g|$ satisfy the following inequalities: $$ \frac{\partial}{\partial r}\sqrt{g_{\theta\theta}(r,\theta)}\geq\frac{\partial}{\partial r}\log[\sin(br)/r], \ g_{\theta\theta}(r,\theta)\geq[\sin(br)/br]^2 \ \ (\mbox{if} \ br<\pi) $$ $$ \frac{\partial}{\partial r}\sqrt{g_{\theta\theta}(r,\theta)}\leq\frac{\partial}{\partial r}\log[\sin(ar)/r], \ g_{\theta\theta}(r,\theta)\leq[\sin(ar)/ar]^2 $$ $$ \frac{\partial}{\partial r}\sqrt{|g(r,\theta)|}\leq(n-1)\frac{\partial}{\partial r}\log[\sin(\alpha r)/r]\leq-a'r/3, \sqrt{|g(r,\theta)|}\leq\left[\frac{\sin(\alpha r)}{\alpha r}\right]^{n-1} $$ $$ \frac{\partial}{\partial r}\sqrt{|g(r,\theta)|}\geq(n-1)\frac{\partial}{\partial r}\log[\sin(br)/r], \sqrt{|g(r,\theta)|}\geq\left[\frac{\sin(\alpha r)}{br}\right]^{n-1} \ (\mbox{if} \ br<\pi) $$
I don't understand how to use this theorem to obtain the inequality above. Has anyone studied this demonstration of this book? Thank you.
