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Consider a hypersurface (not necessarily compact) smoothly embedded into $\mathbb{R}^n$ such that the Hessian is a positive definite bilinear form. Due to positivenes, Hessian can be taken as a metric for this hypersurface (call it Hessian metric). On the other hand, since it is an embedding, we have an induced metric (from ambient Euclidean metric) on it. My question would be as follows: is there a way to obtain Hessian metric from an induced metric? Through some kind of deformation, or maybe through some geometric flow (start with induced metric as an initial metric and end up with the Hessian metric)?

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    $\begingroup$ It's not clear what you mean. Are you regarding the hypersurface as a graph and taking the Hessian of the graphing function? If the hypersurface can be written as a graph in more than one way, you'll get different Hessian forms on the hypersurface for different graphical representations, so 'the Hessian metric' is not a well-defined notion. For example, even for the curve $xy=1$ in the first quadrant of the $xy$-plane, the Hessian metric you get for $y=1/x$ is $2x^{-3}dx^2$ while the one for $x = 1/y$ is $2y^{-3}dy^2$, and these are not the same. $\endgroup$
    – Robert Bryant
    Commented Jan 25, 2016 at 9:38
  • $\begingroup$ @RobertBryant Yes, I meant that hypersurface is a graph. The problem is that in certain optimization algorithms one has a manifold over which optimization occures and when hessian is positive definite, one takes it for the role of Riemannian metric. So I started to wonder if one could obtain such metric from an induced one. $\endgroup$
    – Tomas
    Commented Jan 25, 2016 at 10:13
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    $\begingroup$ The point I am making is that, without specifying how the hypersurface is being regarded as a graph, there is no canonical choice of Hessian metric, so there can't be a way to compute it from the induced metric alone, which doesn't depend on how a hypersurface is (or even can be) interpreted as a graph. $\endgroup$
    – Robert Bryant
    Commented Jan 25, 2016 at 12:09
  • $\begingroup$ A simple example is $z = x^2$. The induced metric is the standard flat one, also induced by $z=0$, but the two embeddings have different Hessians. $\endgroup$
    – Deane Yang
    Commented Jan 25, 2016 at 12:33

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