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In simple terms, a manifold is just a generalization (and therefore includes) the simpler geometric spaces you already know, odd shaped spaces if you like, which if you just look at a small piece, are still Euclidean.
What makes Geometry Euclidean. I mean, could you provide me a specific example of a manifold object, or a non-manifold one and give the explicit reasons why they are so? I'm looking fr a comprehensive definition.
Thanks regardless, but i'm really trying to understand this.
*the Geometry of the objects Euclidean
"What makes Geometry Euclidean" Euclidean space is the space we are in and the rules that apply there (usually modelled as R^3). Other sorts of spaces include hyperbolic space or spacetime. Say you take a sphere (I already said it is a generalization and includes the familiar). OK, you know it is a sphere but if you were standing on it (like on the Earth) it would not be immediately obvious that it was anything other than flat (ie Euclidean). Also have a look here http://en.wikipedia.org/wiki/Thurston_conjecture
I understand that but i've heard of several other qualifying "features" that make a figure Manifold. All I understand is that single point, and i've hear other very, very different things, which is why i'm trying to find everything I can to really, really nab it.
(I'm also not sure who's right. I've seen/heard so many different things on this. Also, thanks so much for this stuff so far.)
If you're trying to think about it non-rigorously, the easiest way to think about a manifold is to think about Earth. Earth is a 3-dimensional object, but we live on a 2-dimensional plane on the "outside" of that object. That's the easiest example of a manifold.
If you want more rigorous than that it kinda does require a decent grounding in topology.
If so, then where would I learn more about topology in this sense? When I hear about topology the few times it's been commonly spoken, lol, it's generally referring to less rudimental, fundamental geometrical concepts and more so to maps and geography.
You might be thinking of topography. Topology is a pretty important area of mathematics, if you're looking for an introductory course I think usually recommended is the text Topology by JR Munkres
Topology is generally pretty advanced stuff, though, so keep that in mind going in.
Did you look at the link I gave you? (and the linked Uniformization theorem). Understanding this stuff requires some background in topology and classical geometries plus Riemannian geometry (actually the more geometry/topology the merrier). Instead of asking what is a manifold (which I think I have explained) you need to ask what do I want to do with it? Do you want to do calculus on it, for example. Then you need your manifold to be differentiable.In what dimension do you want to work? 2,3, infinite?
This one? http://www.amazon.com/Topology-First-Course-James-Munkres/dp/0139254951 I looked at the link, but I understood little at first glance. I'm willing to try to read that but it won't make much sense to me at the moment, lol. Yeah, I have no background in any of that, i'm just a curious high school student.
I've done precalc and a very minimal amount of calculus, so, yeah...lol.
Pretty sure that's the right book, yeah. To be honest though, I would wait until you've done a lot more math.
Topology usually isn't even introduced until relatively late in the course of undergraduate studies, and it will be very helpful to have taken some analysis first.
Okay. At what point would you think I would start to comprehend this stuff or at least study it? I'm not going for a math major, but I am doing Physics for my undergrad and might pursue math later at a separate time. Oh, okay.
If you are that way inclined you could try to get to grips with Special Relativity as a way in...
I understand Special Relativity to a decent degree, but I don't see how it could be a way in other than the analogy between the relativity of frames of reference to an object being two dimensional or Manifold, unless that's what it was. Thanks so much man. I would medal both you guys if I could, lol.
I don't remember exactly the ordering of courses in most undergrad math programs, but topology is definitely after a good amount of calculus, differential equations, linear algebra, and i'm pretty sure it comes after real analysis and modern algebra also. It requires a pretty decent mathematical grounding. As for special relativity, I think both special and general relativity rely on topology to a decent degree, but I have not studied them in-depth so that's just a guess.
Yeah. I've only studied the conceptual components of Relativity overall to a basic degree and worked with the already derived equations of stuff so I didn't have to go through the calculus or complex math of any of it. Well, in general, thanks. Looks like i've got a long way to go...
Haha we all have a long way to go, use it as an inspiration :)
Oh, i've seen this before. I've never seen it referred to as that before, though.
Well worth finding a history of the events around that time and the people involved.