Timeline for How do you recover the structure of the upper half plane from its description as a coset space?
Current License: CC BY-SA 2.5
9 events
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| Mar 30, 2010 at 13:56 | history | edited | Charlie Frohman | CC BY-SA 2.5 |
deleted 4 characters in body
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| Mar 29, 2010 at 20:21 | comment | added | Tom Church | In general left-invariant metrics on a Lie group, even a simply connected one, can be non-isometric -- Tam Nguyen Phan showed me a nice example of different left-invariant metrics on S^3. But the pushforward metric on SL(2,R)/SO(2) is homogeneous (acted on transitively by isometries), and thus of constant curvature, and so is either S^2, R^2, or H^2. You can then check (in any number of ways) that it's not flat, so must be H^2. | |
| Mar 29, 2010 at 18:49 | comment | added | Mariano Suárez-Álvarez | A Riemann surface structure is the same thing as a conformal class of metric structures. | |
| Mar 29, 2010 at 18:15 | comment | added | Qiaochu Yuan | Cool. So there remains another question which is maybe better asked as a separate question: what is the connection between the metric structure on H and the Riemann surface structure? | |
| Mar 29, 2010 at 17:26 | comment | added | Mariano Suárez-Álvarez | Quaochu, pick any metric on the Lie algebra---that is, the tangent space at the identity element---and translate it. | |
| Mar 29, 2010 at 17:24 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
Fix a tex display
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| Mar 29, 2010 at 17:24 | history | edited | Douglas Zare | CC BY-SA 2.5 |
Typesetting
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| Mar 29, 2010 at 17:14 | comment | added | Qiaochu Yuan | Thanks for your answer! What's an example of a left-invariant metric on SL_2(R)? Does it matter which one I choose if there are more than one? | |
| Mar 29, 2010 at 17:09 | history | answered | Charlie Frohman | CC BY-SA 2.5 |