Timeline for Is there any intrinsic ( without any reference to embedding ) and coordinate free , basis free definition of a differentiable manifold? [duplicate]
Current License: CC BY-SA 4.0
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| Jan 28, 2019 at 8:31 | history | closed |
Francois Ziegler Wojowu Steven Landsburg Lee Mosher Stefan Waldmann |
Duplicate of Is there a sheaf theoretical characterization of a differentiable manifold? | |
| Jan 28, 2019 at 3:57 | comment | added | Deane Yang | Normally, when we say simply “a manifold”, we don’t assume it to have any geometric structure on it. The sphere, purely as a manifold, has no specific geometric properties. However, the unit sphere in $\mathbb{R}^n$ does. | |
| Jan 27, 2019 at 23:37 | comment | added | Sheldon | How is that not clear to you. I specifically asked for things like gaussian curvature, metric tensor etc..which surely isn't purely topological. PS geometrical and topological aspects of geometric objects are never completely independent.. | |
| Jan 27, 2019 at 22:51 | comment | added | Deane Yang | It's not clear, at least to me, whether you're asking about a way to define, without using coordinates or an embedding into Euclidean space, the topological structure of a smooth manifold or the geometric properties of a Riemannian manifold without using coordinates. A smooth manifold has no geometric properties, only topological ones. In particular, there is no concept of curvature on a manifold per se. However, if the manifold has a Riemannian metric or is emedded into Euclidean space, then the metric or embedding has geometric properties such as curvature. | |
| Jan 27, 2019 at 22:24 | history | edited | Sheldon | CC BY-SA 4.0 |
added 297 characters in body
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| Jan 27, 2019 at 20:15 | comment | added | Steven Landsburg | Anything that involves a preferred class of geodesics is going to have a lot more structure than a differentiable manifold. | |
| Jan 27, 2019 at 17:00 | review | Close votes | |||
| Jan 28, 2019 at 8:31 | |||||
| Jan 27, 2019 at 16:42 | answer | added | Victor Petrov | timeline score: 0 | |
| Jan 27, 2019 at 16:30 | review | First posts | |||
| Jan 27, 2019 at 16:58 | |||||
| Jan 27, 2019 at 16:29 | comment | added | Wojowu | math.stackexchange.com/q/2010035/127263 | |
| Jan 27, 2019 at 16:28 | history | asked | Sheldon | CC BY-SA 4.0 |