# Covariant Derivatives and the Cross Product

I've recently read a paper that used a covariant derivative product rule for cross products. It was something like $\nabla_v (A \times B) = (\nabla_v A) \times B + A \times (\nabla_v B)$. Here, $A, B$ are vector fields on a regular surface $S$, and $\nabla$ is the Levi-Civita connection.

From my limited knowledge of covariant derivatives, this seems implausible. For instance, I believed that the output of the covariant derivative always lies in the tangent plane, which seems to contradict the above rule. For example, assume $A$ and $B$ are tangent vectors, then $\nabla_v A, B$ are tangent vectors, so $\nabla_v A \times B \notin T_p(S)$.

I'd like to read up on this, but none of my standard books on differential geometry cover the cross product. I understand that the cross product is a (1, 2) tensor, so it should follow the product rule associated with tensors, but I'm not sure if that results in the product rule above. Could any differential geometers please give me a reference? Thanks!

• The output of the covariant derivative always lies in the tangent plane only if you take the covariant derivative of the tangential fields w.r.t. to a tangential field. In your case you differentiate something ambient. In fact, $A \times B$ will be normal to the surface. – Yuri Vyatkin Mar 21 '12 at 22:22

One of the quick ways to see that is to recall that the cross product can be equivalently defined by $$A \times B := (A^{\flat} \wedge B^{\flat})^{\sharp}$$
Assuming that the covariant derivative works as expected we may write the following lines \begin{align} \nabla_{v}(A \times B) &= (\nabla_{v}A^{\flat} \wedge B^{\flat} + A^{\flat} \wedge \nabla_{v}B^{\flat})^{\sharp} \\ &= (\nabla_{v}A^{\flat} \wedge B^{\flat})^{\sharp} + A^{\flat} \wedge \nabla_{v}B^{\flat})^{\sharp} \\ &= \nabla_{v}A \times B + A \times \nabla_{v}B \end{align}
If we look at this calculation more carefully we will observe that there are different covariant derivatives that are involved there! The normal connection acts on the normal fileds such as $A \times B$ here, while the "intrinsic" covariant derivative acts on the tangential fields $A$ and $B$.
We should have adorned our $\nabla$-s with some marks to distinguish them with regards to the bundle they act on, but this is quite customary in differential geometry to use the same $\nabla$ for all bundles involved in calculations provided the reader knows where the sections are taken from.