Manifold definition


by TrickyDicky
Tags: definition, manifold
lavinia
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#37
Feb11-13, 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.
micromass
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#38
Feb11-13, 06:25 PM
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Quote Quote by TrickyDicky View Post
So what?. As micromass noticed I was actually demanding smooth charts, can you have a smooth chart on the point of the cone?
Yes. You can give a cone a smooth structure. But you can't make it an immersed submanifold of Euclidean space.
TrickyDicky
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#39
Feb12-13, 04:10 AM
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Quote Quote by micromass View Post
Yes. You can give a cone a smooth structure. But you can't make it an immersed submanifold of Euclidean space.
Ok, so I admit then I don't have a clue how "You can have a chart on the point of a cone" contradicts anything I wrote in my post. Unless Ben was thinking only about cones in R^3, but that defeats the definition of manifold as an intrinsically defined object.
TrickyDicky
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#40
Feb12-13, 04:29 AM
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Quote Quote by lavinia View Post
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.
More specifically in general relativity the domain would be Minkowskian rather than Euclidean. That is basically the content of the Equivalence principle.
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.
Ben Niehoff
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#41
Feb22-13, 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.
TrickyDicky
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#42
Feb22-13, 06:39 PM
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Quote Quote by Ben Niehoff View Post
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.
Thanks Ben, no worries, I probably overreacted a bit.
lavinia
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#43
Feb22-13, 09:26 PM
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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 non-smoothly in another manifold. The cone is a non- differentiable embedding of a disk in three space.


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