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Manifold definition

by TrickyDicky
Tags: definition, manifold
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mathwonk
#19
Feb9-13, 10:23 AM
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i assumed lavinia was thinking of smooth manifolds and thinking of the infinitesimal vector space, i.e. tangent space, structure that gives at a point.
jgens
#20
Feb9-13, 10:27 AM
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Quote Quote by mathwonk View Post
i assumed lavinia was thinking of smooth manifolds and thinking of the infinitesimal vector space, i.e. tangent space, structure that gives at a point.
lavinia specifically mentioned coordinate systems providing a local vector space structure in post 13 (the post that I responded to) which makes me think he/she meant pulling a vector space structure back using charts. He/she is of course right that you can do this, I just wanted to note that this structure will generally not be coordinate independent.
TrickyDicky
#21
Feb9-13, 02:01 PM
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Quote Quote by jgens View Post
lavinia specifically mentioned coordinate systems providing a local vector space structure in post 13 (the post that I responded to) which makes me think he/she meant pulling a vector space structure back using charts. He/she is of course right that you can do this, I just wanted to note that this structure will generally not be coordinate independent.
And I think this is an important point if only because I'd say this is a common source of confusion when going from Euclidean space to general manifolds.
As I see it this lack of a canonical basis for the local charts on the the manifold is precisely what makes the use of connections necessary to relate vectors from different tangent spaces linearizing open neighbourhoods at different points.
TrickyDicky
#22
Feb10-13, 09:10 AM
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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.

Quote Quote by micromass View Post
I guess I don't know why you specifically want a vector space structure to define manifolds.

All you need to define a manifold is the topological space [itex]\mathbb{R}^n[/itex], and not the addition and multiplication structure.
If you want to define a manifold as something locally homeomorphic to a real vector space (with some natural topology), then you are kind of implying that the vector space structure is important when defining manifolds. But in reality, there is no need for any algebraic structure when defining manifolds. So I'm not sure why you want to look at topological vector spaces to begin with.
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.

Quote Quote by Ben Niehoff View Post
what makes a vector space a vector space is its algebraic structure; that elements of the space can be added, subtracted, and mutiplied by scalars.
Such operations do not, however, make sense on a simply-connected open neighborhood of some manifold. Therefore a manifold cannot be locally compared to a vector space in any meaningful way. The tangent space at each point is a vector space, but points on the manifold itself have no algebraic operations defined on them.
Such operations make sense once you have the tangent space at each point and this is made possible thru the requirement of having the neighbourhoods of points homeomorphic to finite dimensional real vector space (with its natural topology of course but this should be understood from the use of the term homeomorphic).
micromass
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Feb11-13, 01:28 AM
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Quote Quote by TrickyDicky View Post
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.


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.
Now you are talking about differentiable manifolds. The linear structure is important there because the linear structure is important to talk about derivatives and integrals.
If we talk about topological manifolds, then the linear structure seems much less important. There is no (useful) analogue for tangent spaces. Furthermore, I rarely need a linear structure when talking about topological manifolds.

Such operations make sense once you have the tangent space at each point and this is made possible thru the requirement of having the neighbourhoods of points homeomorphic to finite dimensional real vector space (with its natural topology of course but this should be understood from the use of the term homeomorphic).
I don't really know how much the neighborhoods of the points matter if you want to define a tangent space. In the philosophy of Noncommutative geometry, you just need a function ring and derivations on it. So the concept of tangent space can be greatly generalized. On the other hand, if you want to see some kind of relation between the tangent space and curves through a point, then you need smooth charts. And this relation is of course of great importance.
TrickyDicky
#24
Feb11-13, 02:45 AM
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Quote Quote by micromass View Post
Now you are talking about differentiable manifolds. The linear structure is important there because the linear structure is important to talk about derivatives and integrals.
If we talk about topological manifolds, then the linear structure seems much less important. There is no (useful) analogue for tangent spaces. Furthermore, I rarely need a linear structure when talking about topological manifolds.



I don't really know how much the neighborhoods of the points matter if you want to define a tangent space. In the philosophy of Noncommutative geometry, you just need a function ring and derivations on it. So the concept of tangent space can be greatly generalized. On the other hand, if you want to see some kind of relation between the tangent space and curves through a point, then you need smooth charts. And this relation is of course of great importance.
Ok, you are right, I should have made that distinction clearer. Point taken, thanks micro.
TrickyDicky
#25
Feb11-13, 04:25 AM
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I would like to add that even though what you are saying is true in general, in special cases that are heavily used in physics and engineering the distinction between topological and differentiable manifolds vanishes in the sense that for low dimensional manifolds(n<4) not only all diffeomorphisms are homeomorphisms like it's always the case, but also all homeomorphisms are diffeomorphisms, there is a unique differentiable structure so that all charts are smooth.
micromass
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Feb11-13, 05:15 AM
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Quote Quote by TrickyDicky View Post
but also all homeomorphisms are diffeomorphisms
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.
WannabeNewton
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Feb11-13, 05:17 AM
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Quote Quote by TrickyDicky View Post
but also all homeomorphisms are diffeomorphisms
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.
TrickyDicky
#28
Feb11-13, 05:36 AM
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Quote Quote by micromass View Post
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.
Sorry. I'm talking about homeomorphic manifolds. Thought it was clear by the context.
micromass
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Feb11-13, 05:39 AM
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Quote Quote by TrickyDicky View Post
Sorry. I'm talking about homeomorphic manifolds. Thought it was clear by the context.
But [itex]\mathbb{R}[/itex] is homeomorphic (and even diffeomorphic) to [itex]\mathbb{R}[/itex]...
TrickyDicky
#30
Feb11-13, 05:46 AM
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Quote Quote by micromass View Post
But [itex]\mathbb{R}[/itex] is homeomorphic (and even diffeomorphic) to [itex]\mathbb{R}[/itex]...
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.
Gee, this is tough and I'm too fast drawing conclusions. I'll try to refrain that.
Ben Niehoff
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Feb11-13, 09:40 AM
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Quote Quote by TrickyDicky View Post
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.
TrickyDicky
#32
Feb11-13, 01:06 PM
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Quote Quote by Ben Niehoff View Post
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?


Quote Quote by Ben Niehoff View Post
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.
Quote Quote by Ben Niehoff View Post
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.
WannabeNewton
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Feb11-13, 02:10 PM
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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.
Ben Niehoff
#34
Feb11-13, 03:12 PM
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Quote Quote by TrickyDicky View Post
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.
TrickyDicky
#35
Feb11-13, 03:52 PM
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Quote Quote by WannabeNewton View Post
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.
TrickyDicky
#36
Feb11-13, 04:05 PM
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Quote Quote by Ben Niehoff View Post
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.


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