Define the Hessian of a differentiable function $F \colon M \to \mathbb{R}$ where $M$ is a Riemannian manifold by $$(\nabla^2 F)(X,Y) = X(Y(F)) - (\nabla_X Y)(F)$$ where $\nabla$ is the connection on $M$.
If $(U, \varphi = (x^1, \dots , x^n)$ is a system of local coordinates in a neighborhood $U$ of $M$, denoting $\nabla^2 F = F_{ij}dx^i dx^j$, how are the components $F_{ij}$ relate to the function $F$ and the Christoffer symbols $\Gamma_{ij}^k$ of the connection?
If we denote $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$, then $$Y(F) = Y^j \frac{\partial F}{\partial x^j}$$ so $$X(Y(F)) = X^i \frac{\partial Y^j}{\partial x^i}\frac{\partial F^2}{\partial x^i \partial x^j}$$ or that's what I think. However, I'm not sure how the expression $(\nabla_X Y)(F)$ translates into local coordinates. Some help would be appreciated. Thank you.