In this lecture we extend the previous results in the framework of smooth manifolds. The main idea to extend those results is that, similar computations may be performed in local coordinates charts and then we use a partition of unity.
Lemma: Let be a paracompact manifold. Let be a locally finite covering of such that each is compact. Then, there exists a system of smooth functions on such that:
- Each has a compact support contained in ,
- , .
We recall that on a topological space , a covering is said to be locally finite if each has a neighborhood that intersects only finitely many of the sets ‘s. The space is said to be paracompact if for each covering of , there is a locally finite covering of which is a refinement of .
From now on, in this lecture will be a smooth manifold with dimension .
Definition: A Riemannian structure on is a smooth, symmetric and positive tensor on .
In other words, a Riemannian structure induces for each an inner product on the tangent space and the dependence is required to be smooth.
Unlike the case of , in general we may not define a Riemannian structure on a manifold by using global frames. For instance on the two-dimensional sphere , it is impossible to find smooth vector fields such that for every , is a basis of . However, of course, we may always deal with local orthonormal frames: That is, if is a smooth Riemannian manifold (i.e. a smooth manifold endowed with a Riemannian structure), for every in , we can find an open set and smooth vector fields on such that for every , is an orthonormal basis of the tangent space for the inner product .
From now on we consider a smooth Riemannian manifold . It is possible to find a locally finite covering of by local coordinate charts and smooth vector fields on such that for every , is an orthonormal basis of the tangent space for the inner product . Let be a partition of unity subordinated to this covering.
Our first goal is to define the canonical Riemannian measure on . The vector fields induce smooth vector fields on . Without loss of generality, we may assume that on , . Consider on the Borel measure with density , where is the Lebesgue measure on . If is a non negative Borel function with a compact support included in , it is natural to define We observe that if the support of is included in , so that is well defined. Now, for a general non negative Borel function , we define where is the partition of unity subordinated to the covering . This defines a Borel measure on which is called the Riemannian measure.
The same idea allows to construct the Laplace-Beltrami operator on . If is a smooth function on , we define where is the smooth vector field on the open set constructed as in the linear case. Let us now observe that, on , we have . This leads to the following definition of the Laplace-Beltrami operator on : If is a smooth function, , where is the partition of unity subordinated to the covering .
Exercise: Show that the Laplace-Beltrami operator is symmetric with respect to the Riemannian measure .
Diffusion operators on manifolds are intrinsically defined as follows:
Definition: Let be the set of smooth functions and be the set of continuous functions . A diffusion operator is an operator
- is linear;
- is a local operator; That is, if coincide on a neighnorhood of , then ;
- If has a local minimum at , .
And it is easily seen, that the Laplace-Beltrami is a diffusion operator. It is moreover elliptic in the sense that if is a local coordinate chart, then the operator read in this chart is an elliptic operator on .
As usual, we associate to the differential bilinear form The bilinear form is related to the notion of Riemannian gradient.
Definition: Let be a smooth function. There is a unique smooth vector field on which is denoted by and that is called the Riemannian gradient that satisfies for every and , , where is the differential of .
If is an open set of , and are smooth vector fields on such that for , is an orthonormal frame of , it is readily checked that
The bilinear form is related to the Riemannian gradient by the following formula:
Lemma: Let be smooth functions. We have
Proof: Let . Let be an open neighborhood of and smooth vector fields on such that for , is an orthonormal frame of . On , we have so that in particular Since is arbitrary, the proof is complete
The last result we wish to extend to manifolds is the relation between completeness and essential self-adjointness of the Laplace-Beltrami operator.
The Riemannian distance on a Riemannian manifold is defined exactly as in .
Given an absolutely continuous curve , we define its Riemannian length by If , let us denote by the set of absolutely continuous curves such that The Riemannian distance between and is then defined by The Hopf-Rinow theorem also holds on manifolds:
Theorem:(Hopf-Rinow theorem on manifolds) The metric space is complete if and only the compact sets are the closed and bounded sets.
We then have the following expected theorem:
Theorem: If the metric space is complete, then the Laplace-Beltrami operator is essentially self-adjoint on the space of smooth and compactly supported functions .