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Covariant derivative

asked 2023-08-29 21:17:39 +0100

Richard_L gravatar image

I understand how to compute the Levi-Civita connection on a pseudo-Riemannian manifold. I would like now to use it to compute the covariant derivative in various directions, as hinted in the code commentary to class Levi-Civita Connections. Is there a already existing method which I am overlooking or must I use explicit sums over coefficient indices?

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answered 2023-08-29 23:02:38 +0100

eric_g gravatar image

updated 2023-08-31 10:17:12 +0100

No, there is no dedicated method to compute the covariant derivative of a tensor field along a given vector field. You have to perform the contraction with the vector field explicitly. Note that you can use the method contract to avoid explicit sums on the indices. For instance, to compute the covariant derivative $\nabla_v T$ of a tensor field $T$ along the vector field $v$, simply write


It would be desirable though to implement a method covariant_derivative to perform this directly, by writing

T.covariant_derivative(v, connection=nabla)

which could be shortened to


when there exists a default connection, e.g. on a pseudo-Riemannian manifold. This would be on the same footing as the existing method for the Lie derivative: T.lie_derivative(v) evaluates $\mathcal{L}_v T$.

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Asked: 2023-08-29 21:17:39 +0100

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Last updated: Aug 31 '23