I'm trying to prove the following:
A critical point of $f \in\mathfrak {F} (M)$ is a point $p \in M $ in which $ df_ {p} = 0.$ Then:
a) at such a point there is a Hessian function $H: T_ {p} (M) \times T_ {p} (M)\rightarrow\mathbb{R}$ such that $H (X_ { p}, Y_ {p}) = X_ {p} (Yf) = Y_ {p} (Xf)$ for all $X, Y \in \mathfrak {X} (M).$
b) $H$ is symmetric, Bilinear and satisfies $H (\partial_ {i} | _ {p}, \partial_ {j} | _ {p}) = (\partial ^ {2 } / \partial x ^ {i} \partial x ^ {j}) (p)$ relative to a coordinate system.
Here $M$ is the differentiable manifold, $\mathfrak {F} (M)$ is the set of all differentiable functions of $M$ in $\mathbb {R}$, $\mathfrak {F} (M)$ corresponds to the set of all vector fields in $M$ and $T_ {p} (M)$ is the set of all vectors tangent to $M$ in $p.$
Assuming part a), the properties of symmetry, bilinearity and equality with the partial derivatives are followed by the definition of vector field and derivation.
The part where I'm stuck is a). How do I prove the existence of such a mapping? My idea is that if $p$ is a critical point, it should be possible, by means of charts of the variety $M$ and of $\mathbb {R}$ 'bring the Hessian matrix' from such a point in $\mathbb {R}$ but I can't get this.
Any help is appreciated in advance.