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,