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Im reading book of Milnor, Morse theory and at the very beginning, he defines the hessian as in the paragraph above.

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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?

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An extension of $v \in T_pM$ to a vector field $V$ is just any $V \in \Gamma(TM)$ such that $V_p = v$. There are many extensions of a given vector, which is why we must show that the result is well-defined; i.e. independent of the extensions we choose.

There is no sensible notion of a "constant" vector field on a general manifold - at best this makes sense in a given coordinate chart, but after changing coordinates the field will generally no longer be constant. Your attempted definition makes no sense as far as I can tell: if $v \in T_p M$ then $f \mapsto V(f)$ is always a vector based at $p$ (this is the only point at which it satisfies the derivation axioms), so this does not define a field. The tangent spaces are all isomorphic, sure, but they are not canonically isomorphic.

Regarding the Hessian in Riemannian geometry, yes, these are related: indeed the Hessian at a critical point $$f_{**}(X,Y) = X_p(Yf)$$ is just the special case of the covariant Hessian $$\nabla^2 f(X,Y) = X(Yf) - df(\nabla_X Y)$$ at a point where $df=0.$

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  • $\begingroup$ Thanks, very helpful explanation! $\endgroup$ Dec 5 '17 at 12:07

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