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?
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
nabla(T).contract(v)
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
T.covariant_derivative(v)
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 $\mathcal{L}_v T$.
Asked: 2023-08-29 21:17:39 +0200
Seen: 113 times
Last updated: Aug 31 '23
Copyright Sage, 2010. Some rights reserved under creative commons license. Content on this site is licensed under a Creative Commons Attribution Share Alike 3.0 license.