Happy New Year 2015!

In the last entry in 2014, I talk about conformal change of the Laplace-Beltrami operator. Given a Riemannian manifold of dimension . We denote a conformal metric of where the function is smooth.

Recall the following formula for the Laplace-Beltrami operator calculated with respect to the metric :

where is the determinant of . Then, it is natural to consider the relation between and in terms of . Recall that by we mean, in local coordinates, the following

hence by taking the inverse, we obtain

Clearly,

hence