Timeline for Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm
Current License: CC BY-SA 3.0
13 events
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| Sep 4, 2011 at 23:40 | history | edited | Analysis Now | CC BY-SA 3.0 |
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| Sep 4, 2011 at 21:20 | comment | added | Deane Yang | As for 2), you'll need to figure out the formula for the Hessian with respect to the co-ordinates you want to use for the hyperbolic metric. I once knew what the outcome is, but not anymore. | |
| Sep 4, 2011 at 20:01 | answer | added | José Figueroa-O'Farrill | timeline score: 7 | |
| Sep 4, 2011 at 19:22 | comment | added | Analysis Now | I meant harmonic maps between manifolds, not harmonic functions. | |
| Sep 4, 2011 at 19:10 | comment | added | Analysis Now | @ Deane Yang : thanks for your note, because otherwise I was going think of Hessian defined by the vector of all the Hessians of the co-ordinate functions. I guess people who deal with Harmonic functions might be dealing with these,but I will try to define the Hessian the way you suggested. But for this moment, this just arises as a subproblem of a problem I am looking at, so I would appreciate , if I know what kind of bounds ( most possibly by functions ) the ordinary second order derivative of $\Phi $ can get if the norm of the above defined ( Poincare ) Hessian is $ < \epsilon $. | |
| Sep 4, 2011 at 18:57 | comment | added | Deane Yang | First, make sure you know how to define the differential of $f$ and note that it defines a map between vector bundles that lie over different manifolds (the domain and range of $f$). Then figure out how to define the covariant derivative of a vector bundle map, where both bundles have connections. The Hessian is then just the covariant derivative of the differential of $f$. | |
| Sep 4, 2011 at 18:55 | comment | added | Deane Yang | Oops. I didn't read the question carefully enough. This is about the Hessian of a map between two Riemannian manifolds. That's a lot trickier to define properly. You can't just use the definition of the Hessian of a real-valued function (as mentioned in 1)) and just use it for a map. Defining the Hessian of a map takes real effort. | |
| Sep 4, 2011 at 17:41 | comment | added | Joseph O'Rourke | I took the liberty of adding a link to the paper. | |
| Sep 4, 2011 at 17:39 | history | edited | Joseph O'Rourke | CC BY-SA 3.0 |
Added link, edited out little spacing errors.; deleted 2 characters in body; added 1 characters in body
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| Sep 4, 2011 at 17:25 | history | edited | Analysis Now | CC BY-SA 3.0 |
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| Sep 4, 2011 at 17:21 | comment | added | Deane Yang | There is only one possible co-ordinate-free definition of the Hessian as a tensor. | |
| Sep 4, 2011 at 17:19 | comment | added | Deane Yang | I'm sorry to be unhelpful, but a rather important skill to develop as a differential geometer is figuring out why apparently different definitions and formulas are either really the same or are actually different. There is nothing particularly complicated about the definition of the Hessian of a scalar function, so this is a good learning exercise. The recommended approach is to first figure out what notation and definitions you like best and then slowly learn how to convert anything someone else writes into your preferred notation and definition. | |
| Sep 4, 2011 at 16:59 | history | asked | Analysis Now | CC BY-SA 3.0 |