Suppose is a domain and
is a hypersurface, where
is a function on
. Define
on
. Then
also can be considered as a function on
. How do we understand
?
Denote for short. If we pull the metric of
back to
, denote as
, then
where and
. Then one can use the local coordinate to calculate
also one can see from another definition of Laplacian
By using the expression of stated above, we can calculate
It follows from the definition of tangential derivative on , see, that
then
where is the mean curvature of the
. Combining all the above calculations,