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Gaussian Curvature of a 2D manifold.

asked 2021-05-05 15:07:55 +0200

JackHughes14 gravatar image

Given a two-dimensional manfiold M, a chart X, a metric and the connection coefficents, is it possible to calculate the Gaussian curvature of this manifold using SageManifolds?

I thought that such an opperation would look something like "K = M.gaussian_curvature()", but I get the following attribute error:

'DifferentiableManifold_with_category' object has no attribute 'gaussian_curvature'.

I'm new to SageManifolds, and I'm a little confused with how to proceed.

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Tip: provide code for defining the "givens": a two-dimensional manfiold M, a chart X, a metric and the connection coefficents.

slelievre gravatar imageslelievre ( 2021-05-05 16:46:30 +0200 )edit

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answered 2021-05-05 17:24:36 +0200

eric_g gravatar image

For 2-dimensional Riemannian manifolds, the Gaussian curvature is obtained as half the Ricci scalar:

K = M.metric().ricci_scalar()/2

(Note that you have to invoke ricci_scalar() on M.metric(), not on M). See the documentation of ricci_scalar() for the example of the 2-sphere.

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answered 2021-05-05 16:58:23 +0200

slelievre gravatar image

For parametrized surfaces, the method is called gauss_curvature, see SageMath documentation on parametrized surfaces.

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Asked: 2021-05-05 15:07:55 +0200

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Last updated: May 05 '21