While teaching myself tensor calculus I have come up with this proof of the 2-D hairy ball theorem. When trying to generalize this proof to higher dimensions I get the term $$\nabla_{[i}(x^i \nabla_{j]}x^j) = (\nabla_{[i}x^i )\nabla_{j]}x^j - 1/2R_{ij} x^i x^j $$
In 2 dimensions $(\nabla_{[i}x^i )\nabla_{j]}x^j$ is, up to a constant, the determinant of the $\nabla_{i}x^j$ matrix. When it is zero this means that the direction $y^i\nabla_{i}x^j$ is independent of $y^i$. Therefore $ \nabla_{[i}x^i \nabla_{j]}x^j = 0 $ for constant norm vector fields on 2-d manifolds, since their covariant derivatives have to be perpendicular to the original field.
- What do these terms mean in higher dimensions?
- When are they everywhere zero or everywhere positive (or negative)?
- Does the dimension, being odd or even, play any role?
- What do their integrals mean?
I am particularly interested in these questions for the case when $x^i$ has constant norm. Also is either of these a fairly common or well known tensor? Does either of them have a name? I am working with a positive definite metric and $\nabla$ is the Levi-Civita connection.