- #26

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De function [itex]f:\mathbb{R}\rightarrow\mathbb{R}:x\rightarrow x^3[/itex] is a homeomorphism and not a diffeomorphism. So the statement is not even true for n=1.but also all homeomorphisms are diffeomorphisms

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- #26

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De function [itex]f:\mathbb{R}\rightarrow\mathbb{R}:x\rightarrow x^3[/itex] is a homeomorphism and not a diffeomorphism. So the statement is not even true for n=1.but also all homeomorphisms are diffeomorphisms

- #27

WannabeNewton

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This is not even remotely true. You can define homeomorphisms between two topological manifolds that have no smooth atlas in which case the notion of being a diffeomorphism doesn't even make sense for the homeomorphism.but also all homeomorphisms are diffeomorphisms

- #28

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Sorry. I'm talking about homeomorphic manifolds. Thought it was clear by the context.De function [itex]f:\mathbb{R}\rightarrow\mathbb{R}:x\rightarrow x^3[/itex] is a homeomorphism and not a diffeomorphism. So the statement is not even true for n=1.

- #29

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But [itex]\mathbb{R}[/itex] is homeomorphic (and even diffeomorphic) to [itex]\mathbb{R}[/itex]...Sorry. I'm talking about homeomorphic manifolds. Thought it was clear by the context.

- #30

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Yes, I realize now what I posted is a non-sequitur, the fact that homeomorphic smooth manifolds with dim. n<4 are also diffeomorphic doesn't imply that there is no distinction between topological and smooth manifolds in those dimensions.But [itex]\mathbb{R}[/itex] is homeomorphic (and even diffeomorphic) to [itex]\mathbb{R}[/itex]...

Gee, this is tough and I'm too fast drawing conclusions. I'll try to refrain that.

- #31

Ben Niehoff

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You can have a chart on the point of a cone.Well it is true that a manifold is first of all a topological space with wichever axioms you consider that topological space to have (here I include being Hausdorff, second countable etc), that's understood. But the key property of manifolds seems to be that they can be given charts(coordinate functions) locally, and this can be seen as the property that they can be linearized at any point (they can be assigned tangent spaces at every point).

so it seems the algebraic structure of vector spaces, namely linearity, has some important role in manifolds.

People are actually trying to help you Tricky, best listen and ask for clarification where you don't understand, rather than jumping down their throats.

- #32

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So what?. As micromass noticed I was actually demanding smooth charts, can you have a smooth chart on the point of the cone?You can have a chart on the point of a cone.

That's exactly what I'm doing, read the posts.People are actually trying to help you Tricky, best listen and ask for clarification where you don't understand,

Hmmm, this inaccurate remark can only be attributed off the top of my head to some unexplainable personal grudge you hold against me. If that is the case I pity you, but there isn't much I can do and I particularly don't care anyway, it does bother me a bit that it might spoil an otherwise peaceful(even cheerful) thread. Hopefully it won't.rather than jumping down their throats.

- #33

WannabeNewton

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- #34

Ben Niehoff

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I was referring to statements like this where you externalize your misunderstandings. Everyone else is "not so on target" rather than you having made a mistake. It makes it annoying to answer your questions.I've been pondering a bit on these replies, at first they seemed to me quite reasonable, right now I feel they were not so on target.

I see in your subsequent responses you acted differently. In the past, you often got belligerent; maybe you're not anymore.

- #35

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You are totally right, thanks WN.

- #36

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Nobody makes you answer my questions, if you feel so annoyed by that just don't do it, but please don't lie saying that I jump down anybody's throat, there is a stretch between that and suggesting an answer might be slightly off target wrt what I was referring to, however wrong I may be. More so when micromass had already assertively pointed me to where my confusion was.I was referring to statements like this where you externalize your misunderstandings. Everyone else is "not so on target" rather than you having made a mistake. It makes it annoying to answer your questions.

I see in your subsequent responses you acted differently. In the past, you often got belligerent; maybe you're not anymore.

And your tone is so patronizing, please quit it.

Having said that I have always praised you as an expert in differential geometry in these forums so I encourage you to keep helping people around here.

- #37

lavinia

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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.

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.

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- #38

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Yes. You can give a cone a smooth structure. But you can't make it an immersed submanifold of Euclidean space.So what?. As micromass noticed I was actually demanding smooth charts, can you have a smooth chart on the point of the cone?

- #39

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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.Yes. You can give a cone a smooth structure. But you can't make it an immersed submanifold of Euclidean space.

- #40

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More specifically in general relativity the domain would be Minkowskian rather than Euclidean. That is basically the content of the Equivalence principle.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.

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

Ben Niehoff

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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

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Thanks Ben, no worries, I probably overreacted a bit.

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.

- #43

lavinia

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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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