For functions on $\mathbb{R}^2$, it lets you classify extreme points. When function is log-likelihood, it gives volume of the region where maximum likelihood estimates tend to fall. Are there other places it comes up?
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Yep. The condition that the Hessian is nonsingular at every critical point means that the function in question is suitably "nice," and is called a Morse function. Morse functions are useful because their inverse images $f^{-1}((-\infty, a])$ behave very nicely. Basically, as $a$ varies, you get things that are (up to homotopy equivalence) obtained by attaching cells when $a$ passes through a critical value (the type of cell being determined by the type of the critical point). So since every compact manifold admits a Morse function (the "typical" function is, in fact, Morse--I believe this is a standard transversality idea in differential topology), it follows that every compact manifold is homotopy equivalent to a CW complex. |
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According to Do Carmo, exercise 22 in page 173, the Hessian gives you the second fundamental form of a surface, as John pointed out. Precisely, let $S$ be a surface, $p\in S$ a point and $h: S \longrightarrow \mathbb{R}$ be the height function of $S$ relative to $T_p S$; that is, $h(q) = (q-p)\cdot N(p)$, where $N(p)$ is the normal vector to the surface $S$ at the point $p$. Then the Hessian of the function $h$ at $p$ is the second fundamental form of $S$ at $p$. |
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