# On a Riemannian $3$-manifold, what is the relation between the Levi-Civita connection $\nabla$ and the cross product $\times$?

Let $(M,g)$ be a Riemannian $3$-manifold and $X\perp Y\in \mathfrak{X}(M)$. then

What is the relation between $\nabla$ and $\times$?

Is it possible to calculate $\nabla _X(X\times Y)$ and $\nabla _{(X\times Y)}X$ in general in term of $X,Y$? where $\nabla$ is Levi-civita connection and $\times$ cross product.

## 1 Answer

NB in order for $\times$ to be defined, $M$ must also be oriented.

Hint We may view $\times$ as a $(2, 1)$ tensor, given (pointwise) by $$\times(X, Y) := X \times Y = \operatorname{vol}(X, Y, \cdot)^{\sharp} ,$$ where $\operatorname{vol}$ denotes the Riemannian volume form determined by $g$ (and its orientation) and $\cdot^{\sharp}$ denotes raising an index with $g$. In index notation it is even clearer that $\times$ is just the volume form rewritten using $g$: $\times_{ab}{}^c = \operatorname{vol}_{abd} g^{dc}$. On the other hand, the Levi-Civita connection $\nabla$ preserves both $g$ and $\operatorname{vol}$.