Comparing Lie & Covariant Derivatives of Vector Fields

[kent davidge]

I subtracted the $\mu$-th component of the Lie Derivative of a Vector $U$ along a vector $V$ from the $\mu$-th component of the Covariant derivative of the same vector $U$ along the same vector $V$ and I got $(\nabla_V U)^\mu - (\mathcal{L}_V U)^\mu = U^\nu \partial_\nu V^\mu - V^\nu U^\sigma \Gamma^\mu{}_{\nu \sigma}$

I know I should really say vector field in the above instead of vector. My question is if it's legitimate to perform such subtraction. If so, One notices that the two derivatives are the same when the basis and the vector field $V$ are constant.

 

[haushofer]

Why not? The result is again a tensor.

 

[Orodruin]

As already said, the two terms are both tensors of the same type so it does indeed make sense to take the difference in the sense that the result is a tensor of the same type. However, the more common construction is
$$
\nabla_U V - \nabla_V U - [U,V],
$$
i.e., the torsion acting on U and V, which has a geometrical interpretation.

Also note that the concept of ”constant” vector fields is not well defined on a general manifold.