Lecture 9. Laplace-Beltrami operators on Rn

In this lecture we define Riemannian structures and corresponding LaplaceBeltrami operators. We first study Riemannian structures on Rn to avoid technicalities in the presentation of the main ideas and then, in a later lecture, will extend our results to the manifold case.

We start with the following basic definition:

Definition: A Riemannian structure on \mathbb{R}^n is a smooth map g from \mathbb{R}^n to the set of symmetric positive matrices.

In other words, a Riemannian structure induces at each point x \in \mathbb{R}^n an inner product g_x, and the dependence x \rightarrow g_x is asked to be smooth.

A natural way to define Riemannian structures, is to start from a family of smooth vector fields V_1,\cdots, V_n such that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n. It is then easily seen that there is a unique Riemannian structure on \mathbb{R}^n that makes for x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) an orthonormal basis.

Conversely, given a Riemannian structure on \mathbb{R}^n, it is possible to find smooth vector fields V_1,\cdots, V_n on \mathbb{R}^n such that for x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) an orthonormal basis for this Riemannian structure.

In this course, we shall mainly deal with such a point of view on Riemannian structures and use as much as possible the language of vector fields. This point of view is not restrictive and will allow more easy extensions to the sub-Riemannian case in a later part of the course.

Let us consider a family of smooth vector fields V_1,\cdots, V_n such that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n. Without loss of generality we may assume that \mathbf{det} ( V_1(x),\cdots, V_n(x)) > 0. Our goal is to associate to this Riemannian structure a canonical diffusion operator.

As a first step, we associate with the vector fields V_1,\cdots, V_n a natural Borel measure \mu which is the measure with density d\mu =\frac{1}{ \mathbf{det} ( V_1(x),\cdots, V_n(x))} dx with respect to the Lebesgue measure. This is the so-called Riemannian measure. The diffusion operator we want to consider shall be symmetric with respect to this measure.

Remark: Let (U_1,\cdots,U_n) be another system of smooth vector fields on \mathbb{R}^n such that for every x \in \mathbb{R}^n, (U_1(x),\cdots,U_n(x)) is an orthonormal basis with respect to the inner product g_x. The systems of vector fields (U_1,\cdots,U_n) and (V_1,\cdots,V_n) are related one to each other through an orthogonal mapping. This implies that |\mathbf{det} ( V_1(x),\cdots, V_n(x))|=| \mathbf{det} ( U_1(x),\cdots, U_n(x))|. In other words, the Riemannian measure \mu only depends on the Riemannian structure g.

Due to the fact that for every x \in \mathbb{R}^n, (V_1(x),\cdots, V_n(x)) is a basis of \mathbb{R}^n, we may find smooth functions \omega^{k}_{ij}‘s on $\mathbb{R}^n$ such that [V_i,V_j]=\sum_{k=1}^n \omega_{ij}^k V_k. Those functions are called the structure constants of the Riemannian structure. Every relevant geometric quantities may be expressed in terms of these functions. Of course, these functions depend on the choice of the vector fields $V_i$’s and thus are not Riemannian invariants, but several combinations of them, like curvature quantities, are Riemannian invariants.

The following proposition expresses the formal adjoint with respect to the Lebesgue measure of the vector field V_i which is seen as an operator acting on the spave of smooth and compactly supported functions.

Proposition: If f,g\in \mathcal{C}_c(\mathbb{R}^n, \mathbb{R}) are smooth and compactly supported functions, then we have \int_{\mathbb{R}^n} (V_i f ) g d\mu =\int_{\mathbb{R}^n} f (V_i^* g) d\mu, where V_i^*=-V_i+ \sum_{k=1}^n \omega_{ik}^k.

Proof: Let us denote V_i=\sum_{j=1}^n v_i^j \frac{\partial}{\partial x_j}, and m(x)= \mathbf{det} ( V_1(x),\cdots, V_n(x)). We have \int_{\mathbb{R}^n} (V_i f ) g d\mu =\int_{\mathbb{R}^n}\sum_{j=1}^n v_i^j \frac{\partial f}{\partial x_j} g   \frac{dx}{m} =-\sum_{j=1}^n\int_{\mathbb{R}^n} f \left( m \frac{\partial}{\partial x_j} \frac{1}{m} g  v_i^j \right) d\mu. We now compute \sum_{j=1}^n m \frac{\partial}{\partial x_j} \frac{1}{m}   v_i^j=-\sum_{j=1}^n v_i^j \frac{1}{m} \frac{\partial m}{\partial x_j} +\sum_{j=1}^n\frac{\partial v_i^j}{\partial x_j}. We then observe that \frac{\partial m}{\partial x_j}  = \frac{\partial}{\partial x_j} \mathbf{det} ( V_1(x),\cdots, V_n(x))  =\sum_{k=1}^n \mathbf{det} \left( V_1(x),\cdots,\frac{\partial V_k}{\partial x_j}(x),\cdots, V_n(x)\right).
Thus, we obtain
-\sum_{j=1}^n v_i^j  \frac{\partial m}{\partial x_j} +\sum_{j=1}^n m \frac{\partial v_i^j}{\partial x_j}  =  \sum_{k=1}^n \mathbf{det} \left( V_1(x),\cdots,-\sum_{j=1}^n v_i^j \frac{\partial V_k}{\partial x_j}(x),\cdots, V_n(x)\right)
+\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,\frac{\partial v_i^j}{\partial x_j} V_k(x)  ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,[V_i,V_k](x)  ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n\mathbf{det} \left( V_1(x),\cdots,\sum_{j=1}^n \omega_{ik}^j(x) V_j(x)  ,\cdots, V_n(x)\right)
=-\sum_{k=1}^n \omega_{ik}^k (x)\mathbf{det} \left( V_1(x),\cdots,V_k(x)  ,\cdots, V_n(x)\right)=-m\sum_{k=1}^n \omega_{ik}^k. \square

With this integration by parts formula in hands we are led to the following natural definition
Definition: The diffusion operator L =-\sum_{i=1}^n V_i^* V_i =\sum_{i=1}^n V_i^2 -\sum_{i,k}^n \omega_{ik}^k V_i is called the Laplace-Beltrami operator associated with the Riemannian structure g.

The following straightforward properties of L are let as an exercise to the reader:

  • L is an elliptic operator;
  • The Riemannian measure \mu is invariant for L;
  • The operator L is symmetric with respect to \mu.

Exercise: Let (U_1,\cdots,U_n) be another system of smooth vector fields on \mathbb{R}^n such that for every x \in \mathbb{R}^n, (U_1(x),\cdots,U_n(x)) is an orthonormal basis with respect to the inner product g_x. Show that \sum_{i=1}^n V_i^* V_i=\sum_{i=1}^n U_i^* U_i. In other words, the Laplace-Beltrami operator is a Riemannian invariant: It only depends on the Riemannian structure g.

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