The completion of $C_c^\infty(M)$ with respect to $\lVert \nabla u\rVert_{L^2(M)}$ on a compact Riemannian manifold Let $M$ be a compact smooth Riemannian manifold (eg. a smooth hypersurface) and let $X$ be the space given as the completion of $C_c^\infty(M)$ with respect to the norm $\left(\int_{M} |\nabla u|^2\right)^{\frac 12}.$
Is this space well-defined and sensible? What is it used for?  In particular I am interested about the links to using Poincare's equality. This space surely is not the same as $H^1(M)$ since we are neglecting the $L^2$ norm of $u$ in the definition.
If $M$ were an open domain in Euclidean space, I am told that this space corresponds to "choosing Dirichlet conditions". I think in this case $X=H^1_0$ so that kinda makes sense.
 A: The result of completion is called homogeneous Sobolev space and is denoted $\dot H^1$. In general, dot over the name of a space indicates homogeneous version, meaning the space is equipped with seminorm that takes only one order of derivative into account. The fact that we have a seminorm that not a norm can be a disadvantage sometimes; but on the other hand, the seminorm has nice scaling and invariance properties. Under scaling of the independent variable, the seminorm of $\dot H^1$ is multiplied by $\tau^{n-2}$ where $n$ is the dimension of the space. In particular, in two dimensions the seminorm is scaling invariant, and even conformally invariant: $\|u\circ f\| = \|u\|$ for every conformal map $f$. In contrast, the full $H^1$ norm is not conformally invariant: the $\int u^2$ terms is not preserved by composition with $f$. 
The seminorm is also what appears on the right side of Sobolev and Poincaré inequalities. For example: for every $u\in \dot H^1(M)$ we have 
$$\int_M |u-\bar u|^2\le C\int_M |\nabla u|^2$$
where $\bar u$ is the mean value of $u$ on $M$, i.e., $\int_M u$ divided by the volume of $M$.
Search for "homogeneous Sobolev space" to find more examples of where this concept shows up.
