Tensor Calculus Questions

[johnybristow1]

Let $M$ be a smooth manifold with a non-degenerate metric tensor $g$, and let $X$ and $Y$ be two smooth vector fields on $M$. Define the tensor field $F$ on $M$ by: $$ F(X,Y) = \nabla_XY - \nabla_YX - [X,Y], $$ where $\nabla$ is the Levi-Civita connection of $g$ and $[X,Y]$ is the Lie bracket of the vector fields $X$ and $Y$.