Timeline for Reference for shortest educational path to (Riemannian) hyperbolic plane
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Sep 20, 2022 at 20:33 | history | edited | David Steinberg | CC BY-SA 4.0 |
added 66 characters in body
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| Jan 12, 2022 at 5:04 | vote | accept | David Steinberg | ||
| Jan 10, 2022 at 17:49 | comment | added | David Steinberg | @Bumblebee Thanks! I teach a sort-of-flipped class out of Greenberg's "euclidean and non-euclidean geometry." Happy to share resources if you like, send me an email: dcstein at uoregon dot edu | |
| Jan 8, 2022 at 23:26 | comment | added | Deane Yang | I think the best way to do this is to introduce Minkowski space, show what it means for a surface to be space-like, show that you can still define the first and second fundamental forms the same way you do for a surface in Euclidean space, and show that the sphere of radius $-1$ has Gauss curvature $-1$. You can use stereographic projection to get the disk and half-space models and derive all the synthetic geometric properties. This is all beautifully explained in the paper cited by @MoisheKohan, which is also available here: math.ucdavis.edu/~kapovich/RFG/cannon.pdf | |
| Jan 8, 2022 at 23:20 | answer | added | Deane Yang | timeline score: 1 | |
| Jan 8, 2022 at 17:13 | comment | added | Bumblebee | @David: This sounds like a fantastic undergraduate class. Is there any way I can take a look at your lecture notes/materials? | |
| Jan 8, 2022 at 14:49 | answer | added | Danny Ruberman | timeline score: 6 | |
| Jan 8, 2022 at 2:46 | history | became hot network question | |||
| Jan 7, 2022 at 21:05 | comment | added | Gerry Myerson | @Robert, I think what's meant is that hyperbolic geometry is equiconsistent with Euclidean geometry, in the sense that any contradiction in either one would imply the existence of a contradiction in the other. | |
| Jan 7, 2022 at 20:27 | answer | added | coudy | timeline score: 5 | |
| Jan 7, 2022 at 20:19 | comment | added | roy smith | have you looked at the book by Millman and Parker: Geometry, a metric approach with models? (the review by mathwonk is by me.) amazon.com/Geometry-Metric-Approach-Undergraduate-Mathematics/… | |
| Jan 7, 2022 at 19:52 | comment | added | Robert Bryant | I'm not sure what you mean by "hyperbolic geometry is consistent with Euclidean geometry". Euclidean geometry satisfies the parallel postulate, which fails in hyperbolic geometry. Presumably, you mean that hyperbolic geometry satisfies the incidence, betweenness, and congruence axioms that Euclidean geometry does. | |
| Jan 7, 2022 at 19:40 | answer | added | Moishe Kohan | timeline score: 8 | |
| Jan 7, 2022 at 19:32 | comment | added | Anton Petrunin | Check my book anton-petrunin.github.io/birkhoff it might work for you. | |
| Jan 7, 2022 at 19:02 | comment | added | user44143 | The half-plane model ($ds^2=(dx^2+dy^2)/y^2$) and the disc model ($ds^2=(dx^2+dy^2)/(1-x^2-y^2)^2$) both use manifolds which need only one coordinate chart. Can you ask the students to skip the charts and focus on the coordinates in this case? | |
| Jan 7, 2022 at 18:40 | history | asked | David Steinberg | CC BY-SA 4.0 |