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Manifold definition 
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#37
Feb1113, 04:45 PM

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In the theory of relativity there are local coordinate systems where the observer feels that he is in a Euclidean domain. These are so called free float coordinates. Here the observer can imagine that he can extend his local world beyond the confines of his measuring instruments to a vector space. I think these coordinates are in some sense canonical.
On a general manifold there are no canonical coordinates but on a Riemannian manifold one always has Gaussian polar coordinates and on manifolds with different structures e.g. Riemann surfaces( conformal coordinates) one has other natural coordinates. 


#38
Feb1113, 06:25 PM

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#39
Feb1213, 04:10 AM

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#40
Feb1213, 04:29 AM

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Of course in the presence of a (pseudo)Riemannian metric you may have those kinds of natural local coordinates :geodesic (Fermi) normal coordinates, once you have these it is easy to derive polar coordinates, but I guess they rely on the Riemannian metric. 


#41
Feb2213, 05:27 PM

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Tricky, I'm sorry I snapped at you. I've gotten annoyed with you in the past, but I think my previous impression of you is wrong.
As for charts at the point of the cone: You can define polar coordinates that are centered at the point, and therefore cover the neighborhood of the point in a single patch. You can always scale these coordinates so that the point of the cone is homeomorphic to R^n...for a 2d cone, imagine simply projecting down into a plane to give you the homeomorphism. As Micro points out, you can also use this projection into a flat plane to define a smooth structure at the point, but then you don't really have a conical point anymore. 


#42
Feb2213, 06:39 PM

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#43
Feb2213, 09:26 PM

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P: 1,716

One general principal that this discussion of the cone illustrates is that geometry is a structure that is added onto a topological space and a topological space can be given many geometries.
Another is that a smooth manifold may be embedded nonsmoothly in another manifold. The cone is a non differentiable embedding of a disk in three space. 


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