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.


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 $\operatorname{vol}$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.