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