Is It More Precise to Define Manifolds as Topological Vector Spaces?

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Discussion Overview

The discussion revolves around the definition of manifolds, specifically whether it is more precise to define them as topological spaces locally homeomorphic to topological vector spaces rather than to Euclidean spaces. The scope includes theoretical considerations and conceptual clarifications regarding the nature of manifolds and the implications of different definitions.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • Some participants argue that defining manifolds as topological spaces locally homeomorphic to real topological vector spaces could provide more precision than the traditional definition involving Euclidean spaces.
  • Others assert that a real vector space is a purely algebraic object and does not inherently possess a topology, questioning the validity of the proposed definition.
  • It is noted that any finite-dimensional vector space can be given a norm, which generates a topology, leading to the conclusion that finite-dimensional normed vector spaces are homeomorphic to Euclidean spaces.
  • Some participants express confusion over the ambiguity of the term \(\mathbb{R}^n\), indicating that it can refer to different mathematical structures (algebraic, topological, or metric) depending on context.
  • A later reply emphasizes that the definition of a manifold only requires the topological space aspect, suggesting that the algebraic structure of vector spaces is not necessary for defining manifolds.
  • Participants discuss the implications of infinite-dimensional spaces, noting that they can have multiple topologies, complicating the use of the term "homeomorphic." This raises questions about the appropriateness of the proposed definition in those contexts.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether defining manifolds as locally homeomorphic to topological vector spaces is more precise. Multiple competing views remain regarding the necessity and implications of incorporating vector space structures into the definition of manifolds.

Contextual Notes

There are limitations regarding the definitions of vector spaces and topological spaces, as well as the implications of different topologies in infinite-dimensional contexts. The discussion highlights the need for clarity in terminology and the assumptions underlying various definitions.

  • #31
TrickyDicky said:
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.

You can have a chart on the point of a cone.

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.
 
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  • #32
Ben Niehoff said:
You can have a chart on the point of a cone.
So what?. As micromass noticed I was actually demanding smooth charts, can you have a smooth chart on the point of the cone?


Ben Niehoff said:
People are actually trying to help you Tricky, best listen and ask for clarification where you don't understand,
That's exactly what I'm doing, read the posts.
Ben Niehoff said:
rather than jumping down their throats.
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.
 
  • #33
Just be careful in what structures you pre - impose on your manifold. Things that hold for smooth manifolds don't necessarily have to hold for the more general topological manifold. The point is that in field theories like GR or classical field theory, you almost always only deal with smooth manifolds for obvious reasons so the mention of topological manifolds doesn't really come up but that certainly doesn't mean there is a vanishing of the distinction between topological manifolds and smooth manifolds.
 
  • #34
TrickyDicky said:
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 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.
 
  • #35
WannabeNewton said:
Just be careful in what structures you pre - impose on your manifold. Things that hold for smooth manifolds don't necessarily have to hold for the more general topological manifold. The point is that in field theories like GR or classical field theory, you almost always only deal with smooth manifolds for obvious reasons so the mention of topological manifolds doesn't really come up but that certainly doesn't mean there is a vanishing of the distinction between topological manifolds and smooth manifolds.

You are totally right, thanks WN.
 
  • #36
Ben Niehoff said:
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.

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.
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
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.
 
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  • #38
TrickyDicky said:
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.
 
  • #39
micromass said:
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.
 
  • #40
lavinia said:
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.
 
  • #41
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
Ben Niehoff said:
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.
 
  • #43
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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