Repeated application of the gradient on a Riemannian manifold While reading about Sobolev spaces on manifolds, I encountered the following notation regarding the norm in $H^k$: $\| \nabla ^k f \|_{L^2}$. There are two questions here:
1) What kind of object id $\nabla ^k f$? Is it a $k$-vector? How is it defined?
2) How is the metric $g$ extended to the space of these objects?
I suspect a strong similarity to differential forms and their norms as defined in Hodge theory, yet there must also be some differences, since forms are a purely differential object, whereas the gradient is a Riemannian one.
 A: A connection on $E$ is a map $\nabla: \Gamma(E) \to \Gamma(E \otimes T^*M)$ satisfying certain conditions. By having it act as the Levi-Civita connection on $T^*M$ you inductively also have connections $\nabla: \Gamma(E \otimes (T^*M)^{\otimes_k}) \to \Gamma(E \otimes (T^*M)^{\otimes_{k+1}})$. $\nabla^k$ means a composition of a bunch of these to get a map $\Gamma(E) \to \Gamma(E \otimes (T^*M)^{\otimes_{k}})$. 
It $E$ and $F$ have fiberwise metrics then $E \otimes F$ canonically carries one, which is how you define the $L^2$ norm. Your case is given by the above after specializing $E = \Bbb R$, the trivial bundle. 
(In particular, if you want to write a Sobolev norm on a Riemannian manifold, just use the Levi-Civita connection; if you want to write a Sobolev norm for sections of a bundle $E$ you need both a Riemannian metric on $M$ and a fiberwise metric and connection on $E$.)
Warning. Do not confuse this with the skew-symmetrized version $d_\nabla: \Omega^k(E) \to \Omega^{k+1}(E)$. This is a different beast, and $d_\nabla^2$ is (multiplication by) a 2-form in $\Omega^2(\text{End}(E))$. In particular if $\nabla$ is a flat connection the "second derivative" with $d_\nabla$ of anything is always zero! Clearly not what we want when we're trying to measure how big higher derivatived are. 
A: I have found the answer in Kobayashi & Nomizu, volume 1, page 124. If $K$ is a tensor of type $(r,s)$, then one may construct a new tensor $\nabla K$ of type $(r, s+1)$, defined by
$$(\nabla K) (X_1, \dots, X_s, Y) = (\nabla _Y K) (X_1, \dots, X_s)$$
and thus define inductively $\nabla ^k K$ as $\nabla (\nabla ^{k-1} K)$.
Choosing now $K$ to be $f$, a tensor of type $(0,0)$, makes clear the notation $\nabla ^k f$ as a $k$-form.
The metric on $k$-forms is obtained by first dualizing (i.e. in local coordinates raising the indices, i.e. taking the inverse of the matrix of $g$) the Riemannian metric $g$ (that measures vectors) in order to obtain a metric $g^*$ on $1$-forms, and then considering $(g^*)^{\otimes k}$, the $k$-th tensor power of $g^*$ (or, more precisely, its restriction to the space of exterior forms).
