Timeline for Flat Riemanniann manifolds
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Apr 15, 2010 at 20:19 | comment | added | Benoît Kloeckner | CAT(0) is a more subtle notion. I guess a comment is too short to answer this, but the wikipedia entry en.wikipedia.org/wiki/CAT(k)_space should give you the basics. For more complete study, there are many books; I personally like the small Lectures on spaces of nonpositive curvature by Ballmann very much. | |
| Apr 15, 2010 at 20:17 | comment | added | Benoît Kloeckner | Well, an infinite line is simply a infinite geodesic, or straight line (flat metrics being locally isomorphc to $\mathbb{R}^n$, a flat manifolds inherits an affine structure so affine line make sense). Similarly, the usual notion of curvature is defined locally in the plane (up to an orientation), so that this notion makes sense in a flat manifold. This should clarify the notion of constant curvature curve; "simple" means that the curve does not cross itself. | |
| Apr 15, 2010 at 19:26 | comment | added | Qfwfq | Also, could I ask you a question: which is the meaning of "CAT(0)"? | |
| Apr 15, 2010 at 19:25 | comment | added | Qfwfq | In your paragraph "Any complete flat [...]", do you also mean that such manifolds can be obtained as quotients by a discrete group? | |
| Apr 15, 2010 at 19:22 | comment | added | Qfwfq | As for your second example, what do you mean by "infinite line" and "simple curves"? Thank you. | |
| Apr 15, 2010 at 19:21 | comment | added | Qfwfq | As for your first example, now I realize that probably a cone with a closed neighbourhood of the vertex chopped off would be an example of a non-planar thing obtained "as an open in a quotient". | |
| Apr 15, 2010 at 19:06 | history | answered | Benoît Kloeckner | CC BY-SA 2.5 |