Calculating Curvature of 3D Hyperboloid - Parameters & Extrinsic Curvature

[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

 

[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).

 

[AWA]

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).

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 want to try.
Thanks.

 

[adriank]

In general the intrinsic curvature of a Riemannian manifold at a point isn't a single number, but in general a tensor (called the Riemann curvature tensor). It just happens that for 2D surfaces, this tensor only has one independent component, which is related to the Gaussian curvature. But for 3D manifolds, there are 6 independent components (although you can reduce this down to the Ricci tensor or further to the Ricci scalar, to get a single number. In the 2D case, the Ricci scalar is twice the Gaussian curvature.)

I suspect that you could calculate several radii of curvature to obtain the Ricci scalar. (I don't know any details.)

 

[blue_raver22]

for your question! The curvature of a 3D hyperboloid can be calculated using various mathematical methods, depending on the specific parameters and properties of the hyperboloid. In general, the intrinsic curvature of a surface refers to the curvature that is determined solely by the shape and geometric properties of the surface itself, without reference to any external space. In contrast, the extrinsic curvature of a surface refers to the curvature as seen from an external space, such as a higher dimensional space in your example.

To calculate the intrinsic curvature of a 3D hyperboloid, you will need to know the specific equation or parametric representation of the hyperboloid. This will allow you to calculate the first and second fundamental forms, which are mathematical quantities that describe the intrinsic curvature of a surface. From these fundamental forms, you can then calculate the Gaussian and mean curvatures, which are measures of the intrinsic curvature at a specific point on the hyperboloid.

To calculate the extrinsic curvature of a 3D hyperboloid as seen from a 4D space, you will indeed need to know the curvature radius. This can be obtained from the equation or parametric representation of the hyperboloid and will allow you to calculate the extrinsic curvature in terms of the curvature radius. Other parameters, such as the first and second fundamental forms, may also be needed depending on the specific method used to calculate the extrinsic curvature.

In summary, the calculation of curvature for a 3D hyperboloid involves both intrinsic and extrinsic components and requires knowledge of the specific parameters and properties of the hyperboloid. I hope this helps clarify the process and please let me know if you have any further questions.