Let (M,g) be a riemannian manifold with Levi-Civita connection ∇, and f is a function. I define a Laplacian by the following formula:
Δg(f)=n∑i=1g(∇ei(df∗),ei)
where ei is an orthonormal basis of TM.
What are the proper values of the sodefined Laplacian?
What is the regularized determinant of the Laplacian?