Fractional Sobolev spaces on closed manifolds

Question

Let $M$ be a closed manifold and $0<s<1$. How is the fractional Sobolev space , $H^s(M)$ defined? In particular if $M$ is a closed smooth simple curve in the place how is $H^{1/2}(M)$ characterized?

Answer 1

Pick a finite open cover of charts $\{U_j\}_{j=1}^N$ with diffeomorphisms $s_j: \Bbb R^n \to U_j$. Let $\varphi_j$ be a partition of unity subordinate to the $U_j$. We put a norm on $C^\infty(M)$ as follows:

$$\|u\|_{H^s}^2 = \sum_{j=1}^N \|\varphi_j(s_j(x))u(s_j(x))\|_{H^s}^2,$$ where the norm in the sum is the $H^s$ norm on $\Bbb R^n$.

The point being that we cut a function into parts supported in each chart and then take the Sobolev norm for each part, and sum them up. The norm you get will depend on the cover of charts and partition of unity you choose, but changing these will get you an equivalent norm; this follows from the diffeomorphism invariance of Sobolev spaces.

This defines a norm on $C^\infty(M)$; its completion is the Sobolev space $H^s(M)$.