Let be an -dimensional differentiable connected Riemannian manifold with metric tensor . In order to distinguish from other metrics on , we shall denote the manifold with by .
We start with the following terminology called “a conformal change“.
Definition. If is another metric on , and there is a function on such that , then and are said to be conformally related or conformal to each other, and such a change of metric is called a conformal change of Riemannian metric.
We are now in a position to define the so-called “conformal invariant operators“.
Definition. We call a metrically defined operator conformally covariant of bi-degree if under the conformal change of metric , the pair of corresponding operators and are related by
Example 1. The Laplace-Beltrami operator is conformal invariant.
Proof. Let us recall from this topic that on , the Laplace-Beltrami operator is defined to be
and in local coordinates it is given as follows
By a change of metric we need to calculate and then compare with .
Indeed, in local coordinates
Since the tangent vectors are not affected by conformal change, the rest doesn’t change. Therefore
Example 2. The conformal Laplacian operator is conformal invariant.
Proof. Let us define the so-called conformal Laplacian operator acting on a smooth function as the following
where is nothing but the scalar curvature with respect to metric . We are going to prove under a conformal change , the following identity
holds. Thus if we write
we then have
It is now clear to see that under the conformal change
We know from this topic that under a conformal change
By using example 1, we firstly have
Next we have
which shows that