How is the metric tensor related to the Hessian of the first fundamental form? I know that the metric tensor can not always be formulated as a Hessian, but sometimes it can.  Can you help me to understand what the special conditions are under which the metric tensor is a Hessian of the first fundamental form?  (Note: Einstein notation follows)
If the first fundamental form of some vectors of same length u and v is $$I(u,v) = <u,v> = \vec u^T g \vec v = u^i v^j g_{ij} = u^i v^j <x_i , x_j>, $$ then why doesn't the Hessian of the first fundamental form yield the same quantities as the metric tensor?
$$H_{ij} I(u,v) = \frac{\delta}{\delta x_i} \frac{\delta}{\delta x_j} I(u,v) = u^k v^l \frac{\delta}{\delta x_i} \frac{\delta}{\delta x_j} <x_k , x_l> \ne <x_i , x_j> $$
Can you also provide an equation that shows the relationship between the metric tensor and the Hessian of the first fundamental form?
 A: The first fundamental form is just the metric. If you express it in Riemannian normal coordinates (the Riemannian exponential map centered at $x_0$), the the first derivative of this vanishes at $x_0$, and the Hessian at $x_0$ (which exists since the first derivative vanishes) is the purely covariant form of the Riemann curvature tensor. This is the way how Riemann found his tensor in his price essay for the Paris academy:


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*Commentatio mathematica, qua respondere tentatur quaestioni ab Illma Academia Parisiensi propositae:
``Trouver quel doit être l'état calorifique d'un corps solide homogène indéfeni pour qu'un système de courbes isothermes, à un instant donné, restent isothermes après un temps quelconque, de telle sorte que la température d'un point puisse s'exprimer en fonction du temps et de deux autres variables indépendantes.''
(1861, Gesammelte Mathematische Werke, Zweite Auflage, 391-404). (pdf)
A very good explanation in English of Riemann's work is contained in:


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*Michael Spival: A comprehensive introduction to differential geometry, Volume 2. Publish or Perish, Inc.

A: I'm not sure what you mean by $x^i$ when you write $g_{ij}x^ix^j$. If they are local co-ordinates, then this quantity is co-ordinate-dependent and has no geometric meaning or name. If on the other hand, they are the components of a tangent vector, this quadratic polynomial makes sense and defines a function on the tangent bundle. This function is indeed the metric itself and could be called the first fundamental form.
