Understanding iterated covariant derivatives to define Sobolev spaces on manifolds

Solution 1:

Let $(M,g)$ be a Riemannian manifold with associated Levi-Civita connection $\nabla$. Typically one introduces the covariant derivative on vector fields, i.e., $\nabla:\Gamma(TM)\times\Gamma(TM)\to\Gamma(TM)$, by $(X,Y)\mapsto\nabla_XY$ which satisfies the Leibniz property and some linearity conditions. Any connection on $TM$ uniquely determines a connection on each tensor bundle $T^{(k,l)}M$. In particular, if $u\in C^\infty(M)$, then $$\nabla u(X)=du(X)=X[u],$$ and in coordinates we have that $$\nabla u=\frac{\partial u}{\partial x^j}dx^j,$$ and hence $$g(\nabla u,\nabla u)=g^{ij}\frac{\partial u}{\partial x^i}\frac{\partial u}{\partial x^j}.$$

Given a $1$-form $\omega\in T^{(0,1)}M$, since our connection has generalized, we have that $\nabla\omega\in T^{(0,2)}M$, and in coordinates $$(\nabla\omega)_{ij}=\frac{\partial\omega_i}{\partial x^j}-\Gamma_{ij}^k\omega_k,$$ and so $$g(\nabla\omega,\nabla\omega)=g^{ij}g^{lm}(\nabla\omega)_{il}(\nabla\omega)_{jm}.$$ Letting $\omega=\nabla u$, we then see that $$(\nabla^2u)_{ij}=\frac{\partial ^2u}{\partial x^j\partial x^i}-\Gamma_{ij}^k\frac{\partial u}{\partial x^k}.$$

Let's do this one more time: Given a $(0,2)$-tensor $\omega\in T^{(0,2)}M$, we then have $\nabla\omega\in T^{(0,3)}M$, and $$(\nabla\omega)_{ijk}=\frac{\partial\omega_{ij}}{\partial x^k}-\Gamma_{ij}^l\omega_{lk}-\Gamma_{ij}^m\omega_{km}.$$ Then $$g(\nabla\omega,\nabla\omega)=g^{i_1j_1}g^{i_2j_2}g^{i_3j_3}(\nabla\omega)_{i_1i_2i_3}(\nabla\omega)_{j_1j_2j_3}.$$ Letting $\omega=\nabla^2u$, we then see that $$(\nabla^3 u)_{ijk}=\frac{\partial^3u}{\partial x^k\partial x^j\partial x^i}-\Gamma_{ij}^l\frac{\partial^2u}{\partial x^l\partial x^k}-\Gamma_{ij}^m\frac{\partial^2u}{\partial x^k\partial x^m}.$$

I believe the continued generalization to arbitrary $(k,l)$-tensor fields should be clear from here. Hopefully this helped any confusion.

Edit: For another exposition of the material, cf. Emmanuel Hebby's text "Sobolev Spaces on Riemannian Manifolds".