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
With all these ingredient, we easily obtain
Thus, conformal change of the Laplace-Beltrami operator follows the following rule
When , we simple have
Keep in mind that can be calculated using ; hence one can use conformal changes for and to derive the above formula.
In local coordinates,