## Stereographic projection from center

Suppose we are doing stereographic projection at the center. Namely consider the following map

where . Then one can see , . Under this stereographic projection, a strip will be equivalent to some lens domain on the sphere.

Let us pull the standard metric of to the . For the following statement, we will always omit and . Calculation shows,

therefore

here we used the fact that . Suppose is the connection on equipped with the standard metric. We want to calculate . To that end, it is better to use the coordinates in

Since we know

where means the tangential part to and is the unit normal to . Using the connection in , we get

One can verify from the above equalities that

Similarly

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