Sobolev spaces exponents on Riemannian manifold

Question

I am reading the book of Hebey Non linear Analysis on Manifolds: Sobolev spaces and inequalities, and I am trying to understand the definition he gave of Sobolev spaces on a Riemannian manifold $(M,g)$ (page 21). As long as I understand it, given a function $h\in C^{k}(M)$ we can define its covariant derivative of order $k$ to be the $k$-form defined inductively by $$ \nabla^0h:=h, $$ and for $k\geq 1$ $$ (\nabla^k h)_{i_1,\dots,i_k}:=(\nabla_{i_1}h)_{i_2,\dots,i_k}, $$ where with $\nabla_{I}$ I denote the covariant derivative in the $\frac{\partial}{\partial x^I}$ direction. This definition makes sense to me, since $Dh=dh$ and $D^2 h=\nabla dh$ which is the classical definition of the Hessian. The problem is the exponent in the definition of the Sobolev $W^{k,p}$ norm of $h$: in this book it is given by $$ \int_M g(\nabla^kh,\nabla^k h)^p\,dvol. $$ Here I am trying to use a more compact notation, extending $g$ on the tensors bundle by letting $$ g(dx^i,dx^j):=g^{ij}, $$ and extending this by linearity, setting naturally $$ g(S\otimes T,G\otimes Q):=g(S,G)g(T,Q). $$ What I don't understand is the $p$ exponent: for instance, if $p=2$ this norm does not agree with the norm induced by the inner product $$ <h,f>_k:=\int_M g(\nabla^kh,\nabla^k f)\,dvol, $$ even if the author says so in page 22. I am sure I am missing something really obvious.