Timeline for Is there a family of Riemannian manifolds with explicitly solvable geodesics for manifold M?
Current License: CC BY-SA 4.0
10 events
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| Aug 16, 2019 at 18:45 | comment | added | Justin Dieter | I would like the set of metrics to be diverse in that they can approximate arbitrary metrics fairly well (Like a Fourier series or something) or at least approximate a large number of interesting metrics. And by explicit I do mean closed-form in terms of elementary functions. Although really I only need closed-form in terms of functions that there are algorithms to compute (or approximate to an acceptable degree of accuracy) in very efficient time (this is for a deep learning algorithm that has to do this for a very large number of datapoints). | |
| Aug 15, 2019 at 19:44 | comment | added | Ben McKay | There are results of Matveev that prevent integrable metrics on many manifolds, if I remember correctly. | |
| Aug 15, 2019 at 12:03 | comment | added | user44143 | Related: mathoverflow.net/q/37651/44143. If a closed-form metric is required, then the set of Riemannian metrics will not be very diverse, but perhaps it is possible with only closed-form geodesics. | |
| Aug 15, 2019 at 11:52 | comment | added | Thomas Rot | If you only consider smooth metrics the manifold will not be a Hilbert one, but a Frechet one. A trivial example of $N$ you want is $N$ to be a single metric, for which the geodesic equation is explicitely solvable: e.g. the standard flat metric. Outside of this it seems pretty difficult. I might see some one dimensional examples: For example by taking $N$ to be the family of metrics $e^t g$, where $g$ is the flat metric on Euclidean space. | |
| S Aug 15, 2019 at 11:17 | history | suggested | S.Surace |
add relevant tag
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| Aug 15, 2019 at 9:23 | comment | added | S.Surace | What do you mean by explicit? Do you mean closed-form in terms of elementary functions? | |
| Aug 15, 2019 at 9:20 | review | Suggested edits | |||
| S Aug 15, 2019 at 11:17 | |||||
| S Aug 8, 2019 at 22:37 | history | suggested | Ali Taghavi |
I add a tag.
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| Aug 8, 2019 at 22:37 | review | Suggested edits | |||
| S Aug 8, 2019 at 22:37 | |||||
| Aug 8, 2019 at 21:44 | history | asked | Justin Dieter | CC BY-SA 4.0 |