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Category Archives: Hyperbolic Geo
Hypersurface in hyperbolic space and conformal metric on Sphere
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.
Hyperbolic translation in Poincare disc model
Hyperboloid model is hyperquadric in , with
Halfspace model . Denote ,
Poincaré ball model There are some transformation formula between these hyperbolic models, say
On , there are vertical scaling transformations which are isometries of
On , they are called translation in direction through the map
In component,
On ,
In component,