Suppose is a domain in , , where is the standard metric on . One can construct a hypersurface in hyperboloid by

where and . What are the induced metric on ? One can calculate as the following,

Suppose is a unit eigenvector associated to the eigenvalue of . We need to calculate . Firstly it is easy to see

and

Then

So this gives the following formula

where . Suppose is the eigenvector corresponding to and are orthonormal basis, then

where . Recall that Schouten tensor has the following formula under conformal change,

for and . Therefore the eigenvalues of are .

I am using the thesis of Dimas Percy Abanto Silva and some communication with him.

Advertisements