I can't understand the following definition given by Petersen in his Riemannian Geometry book.
Let $M$ be a Riemannian manifold and let $f \colon M \to \mathbb{R}$ be a smooth function. Let $\nabla f$ denotes its gradient. Then the Hessian Hess$f$ is defined as the symmetric $(0,2)$-tensor $\frac{1}{2}L_{\nabla f}g$, where $g$ is the metric on $M$, and $L$ is the Lie derivative.
How can I prove that in $\mathbb{R}^n$ this does coincides with the usual definition?
Moreover the book says also that we can define the Hessian as a self-adjoint $(1,1)$-tensor by $S(X) = \nabla_X \nabla f$ and the two tensors are naturally related by: $$ \text{Hess}f(X,Y) = g(S(X),Y) \qquad \text{for every smooth vector fields $X$, $Y$.}$$ How can I check that?