As a motivation, consider $\Omega \subset \mathbb{R}^n$ open and bounded. Let $k_x:\Omega \to \mathbb{R}$ be defined as $k_x(y)=\exp(-\frac{\Vert x-y\Vert^2}{2\sigma^2})$. One can show that $$\Vert k_x \Vert_{H_r^2(\Omega)}=\sum_{|\alpha|=0}^r\left(\int_{\Omega}|\partial^{\alpha}k_x(y)|^2dy\right)^{1/2}\leq \left(\frac{1}{\sigma^2}\right)^r\cdot C $$ for $\sigma<1$ and some constant $C>0$ only depending on $\Omega$ and $r$.
Instead of an open subset $\Omega$, I want to derive a comparable inequality for $k_x$ defined as above on a compact Riemannian manifold $M$. The definition of Sobolev norms on manifolds I saw was $$\Vert k_x \Vert_{H_r^2(M)}=\sum_{|\alpha|=0}^r\left(\int_{M}|\nabla^{\alpha}k_x(y)|^2d\mu(y)\right)^{1/2}.$$ My problem is, that all facts I found about those covariant derivatives $\nabla^{\alpha}$ are rather abstract and algebraic, and my knowledge in differential geometry is quite limited. I spent some time to read about it in Do Carmo, but this hasn't brought me any closer to a concrete way to derive those integrals. Neither have the books of Hebey, which seem to be the standard literature on Sobolev-Spaces on manifolds.
I was wondering if someone here has done some computation like this before or has a suitable reference.