# 3D Curvature

1. Aug 20, 2010

### AWA

How can I calculate the curvature of a 3D hyperboloid? I mean, what parameters do I need to calculate the intrinsic curvature?
I guess to calculate the extrinsic curvature as seen from a 4D space I would just need a curvature radius, right?

Thanks

2. Aug 20, 2010

### Ben Niehoff

First write down a parametrization to embed the 3-hyperboloid into R^4.

Then calculate the induced metric on the 3-hyperboloid. This is simply the pullback of the standard R^4 metric with respect to the embedding map.

Once you have the induced metric, you can calculate intrinsic curvature in the standard way.

Note: If you're feeling adventurous, you can take a shortcut by calculating the pullback of the R^4 connection to directly get the connection on the 3-hyperboloid. This would be the most efficient way to do it if you are using the Cartan formalism. But you have to remember that when you pull back connections, there is an extra term (similar to transforming a connection under coordinate transformations).

3. Aug 20, 2010

### AWA

Thanks a lot for your response.
I deduce from the above that I asked something whose answer I should have supposed would be way out of my league since I didn't warn that I don't know about differential geometry.
Is there a way to explain it in plain english for someone who just have basic notions of geometry? without words like embed, connection or pullback?
I know it is not likely that it is possible but maybe somebody wanna try.
Thanks.

4. Aug 20, 2010