Why define $(\nabla^2F)(X,Y)=\nabla_X(\nabla_YF)-(\nabla_{\nabla_XY}F)$?

Solution 1:

In general for a $(p,q)$-tensor $A$, one defines $\nabla A$ as a $(p,q+1)$ tensor by $$\nabla A(X, \cdots) = \nabla_XA(\cdots),$$ where $\nabla_X A$ is defined as (for example if $A$ is a $(2,0)$ tensor) $$\nabla_X A(Y, Z) = \nabla_X(A(Y, Z)) - A(\nabla_X Y, Z) - A(Y, \nabla _X Z).$$

So in our case $$\begin{split} \nabla^2 F(X, Y) &= (\nabla \nabla F) (X, Y) \\ &= (\nabla_X \nabla F)(Y) \\ &= \nabla_X(\nabla F(Y)) - \nabla F(\nabla_XY) \\ &= \nabla_X \nabla_Y F - \nabla_{\nabla_XY} F. \end{split}$$

Remark: Indeed $\nabla^2 F$ is symmetric: $$\begin{split} \nabla^2 F(X, Y) - \nabla^2 F(Y, X) &= X(YF) - \nabla_{\nabla_X Y} F - Y(XF) + \nabla_{\nabla_YX} F \\ &= X(YF) - Y(XF) - \nabla_{\nabla_XY - \nabla _YX} F \\ &= [X, Y]F - \nabla_{[X, Y]} F \\ &= 0 . \end{split}$$