Covariant derivative

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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 $\mathcal{L}_v T$.