Timeline for Obtaining Hessian of the embedding from an induced metric
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 25, 2016 at 12:33 | comment | added | Deane Yang | 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. | |
| Jan 25, 2016 at 12:09 | comment | added | Robert Bryant | 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. | |
| Jan 25, 2016 at 10:13 | comment | added | Tomas | @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. | |
| Jan 25, 2016 at 9:38 | comment | added | Robert Bryant | 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. | |
| Jan 25, 2016 at 8:55 | history | asked | Tomas | CC BY-SA 3.0 |