Im reading book of Milnor, Morse theory and at the very beginning, he defines the hessian as in the paragraph above.
My question is, what is the precise definition of extension to a vector fiels of that vector $v\in TM_p$? Is just the constant vector field $V\in\Gamma(TM) $ such that $V_p(f) =v(f) $ for any $p\in M$ and any $f$ differentiable function on $M$? Because for diferent point $q\in M$, $V_q\in TM_q$ and $V_q=v \in TM_p$. I know that no matter the point the tangent space has the same (finite) dimension so they are isomorphic. So in every point $q$ not $p$, $v$ is the corespondent vector through that isomorphism?
As a part two of this question, at my first course of riemann geometry i have seen a definition of hessian involving the canonical L. C. connection. Is there any conection?