Consider the 2-sphere with the standard metric . All conformal diffeomorphism of are composing a suitable isometries of and some , where is the stereographic projection and on . For any conformal metric , define

If is a conformal transformation, then we can find such that , where

here we use the notation .

**Fact:** for any conformal map of , satisfies the following identity

Actually, this identity means has Gaussian curvature , the same as . Upon this fact, we have the invariance of .

**Proposition:** .

*Proof:*

from integration by parts of the middle term. Suppose , then

So

where we have used . Continuing our simplication of ,

by changing variables. So we only need to prove the last term is , which is

integration by parts, this is equivalent to

From , we get

Changing variable by ,

So we only need to justify . As mentioned at the begining, up to some isometry, for some , where is the stereographic projection of north pole. Then , where is the stereographic projection of south pole. Note that

where if . By changing variables

**Remark1:** Under sterographic projection, is isometric to . Then for on

Another point of view is thinking as a diffeomorphism, then is the transformation of corresponding tangent space. One can also get is the above expression.

**Remark2:** Alice Chang, Paul Yang, prescribing curvature on , 1987.

### Like this:

Like Loading...

*Related*