Given a Compact, Riemannian Manifold $(M,g)$, I'm wondering if we can induce an inner product space on $\mathfrak{X}(M)$, the set of smooth vector fields on $M$ that might have some interesting properties. My tentative definition will be:
Given $X,Y\in\mathfrak{X}(M)$ we'll take $$X\cdot Y=\int_M g(X,Y)dvol_{M}$$ Since $X,Y$ are smooth and $M$ is compact, we can see that this product will be defined for any vector-fields on $M$, and that this product will be positive definite, since $g_p$ is positive definite on $T_pM$ for all $p\in M$.
I believe, that in order to make it complete we would have to do quite a bit of extra work and define some kind of completion on this space since vector spaces of differentiable functions don't end up being complete due to the restrictiveness of differentiability. What kind of work would have to be done to turn the underlying structure into something resembling a well-behaved vectorspace?
Would the properties of such a space tell us anything about the underlying structure of $M$? Are there some notions that I missed, ending up making this space ultimately uninteresting?