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!