A Laplacian in riemannian geometry

Question

Let $(M,g)$ be a riemannian manifold with Levi-Civita connection $\nabla$, and $f$ is a function. I define a Laplacian by the following formula:

$$\Delta_g (f)=\sum_{i=1}^n g(\nabla_{e_i} (df^*), e_i)$$

where $e_i$ is an orthonormal basis of $TM$.

What are the proper values of the sodefined Laplacian?

What is the regularized determinant of the Laplacian?