Metric Tensors for 2-Dimensional Spheres and Hyperbolas

[thehangedman]

Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)?

I know that it's coordinate dependent, so suppose you have two coordinates: with one being like "latitude", 0 at the bottom pole, and 2R at the northern pole, and the other being like longitude, 0 on 1 meridian and Pi * R on the opposite side (here, 2 Pi R gives you the same location as 0).

I've searched online and can't find a simple example of this basic metric tensor... :-(

The other one I'm curious about is the surface of a hyperbola (again, think 2-D surface of a shape in 3 dimensions). What is the metric on THAT surface?

Any type of help is greatly appreciated...

 

[George Jones]

Does anyone know what the metric tensor looks like for a 2 dimensional sphere (surface of the sphere)?

The standard metric is
$$ds^2 = R^2 \left( d\theta^2 + sin^2\theta d\phi^2 \right).$$

 

[HallsofIvy]

Note: George Jones is using the physics notation which takes $\phi$ as the "longitude" and $\theta$ as "co-latitude", the opposite of mathematics notation.