Let me provide another proof of Example 2 in this topic.
Example 2. The conformal Laplacian operator acting on a smooth function as the following
is conformal invariant.
The proof relies on the following fact
Proposition. Under a conformal change , we have
where and are the volume elements with respect to metrics and respectively.
We are now in a position to prove example 2. For simplicity, let us consider the following conformal change . The proof is divided into three steps.
Step 1. Showing
Proof. Indeed, for any test function we have
We now use Laplace-de Rham operator for the sake of simplicity. Note that Laplace–de Rham operator is the Laplacian differential operator on sections of the bundle of differential forms on a pseudo-Riemannian manifold. However, the Laplace-de Rham operator is equivalent to the definition of the Laplace–Beltrami operator when acting on a scalar function. Precisely,
Step 2. Showing the scalar curvature equation
Step 3. Showing