While reading through a set of lectures on the Laplacian on manifolds, I encountered two descriptions of Sobolev spaces.
The first one, valid only for compact manifolds (because it needs to globalize a result using partitions of unity) claims that the Sobolev space $H^s$ can be defined as the completion in $L^2$ of $\mathcal C ^\infty$ under the norm $\| f \| = \left(\sum \limits _i \| (p_i f) \circ h_i ^{-1} \| ^2 _{H^s (U_i)}\right) ^\frac 1 2$, where $\{ (U_i, h_i) \mid i \in I\}$ is an atlas and $\{ p_i \mid i \in I\}$ is a subordinated partition of unity.
The second one, which works only for natural orders $k$, presents $H^k$ as the completion in $L^2$ of $\mathcal C ^\infty$ under the norm $\| f \| _k = \left( \| f \| _{k-1} ^2 + \| \nabla ^k f \| ^2 \right) ^\frac 1 2$. (This description is, in turn, shown to be equivalent to a third one that uses the eigenvalues of the Laplacian.)
My question is: do the first two constructions above produce the same space?
To make things worse, Grigoryan defines the same spaces slightly differently: only for even orders, $W^k = \{ u \in \mathcal D ' \mid u, \Delta u, \dots, \Delta ^k u \in L^2\}$. Is this yet another space?
Why isn't $H^k$ defined simply as $\{ u \in L^2 \mid X_1 \dots X_i u \in L^2 \forall i \le k \forall X_j \in \Gamma(TM) \}$?