What are Manifolds? Understanding the Continuum in Mathematics and Physics

[captain]

what exactly are manifolds? I looked on wikipedia and I am getting the sense that its like n dimensional surface if that makes any sense.

 

[pmb_phy]

what exactly are manifolds? I looked on wikipedia and I am getting the sense that its like n dimensional surface if that makes any sense.

See http://www.geocities.com/physics_world/gr_ma/introduction_to_manifolds.htm

Pete

 

[princeton118]

what exactly are manifolds? I looked on wikipedia and I am getting the sense that its like n dimensional surface if that makes any sense.

Such an abstract concept! such a tough question!
The explanation of manifold on mathematics is very hard to understand! In my opinion, it is just a space which has been defined some properties, which can be used to simulate our spacetime.

 

[CompuChip]

You are probably used to define points in $\mathbf{R}^n$ by specifying their coordinates (with respect to some basis you agreed upon with the one you give the coordinates, so both will have the same point in mind for the same set of numbers).

Manifolds are sort of a generalization to arbitrary spaces. Technically, a manifold is defined as something that looks locally like a Euclidean space $\mathbf{R}^n$ (for some n). The "looks like" is a bijection, so we can go from the manifold to the Euclidean space and back in a unique manner. We just put a coordinate system on the latter in the usual way, and because of the correspondence, we can also consider this as a coordinate system on the manifold (in fact, practically we often forget about the Euclidean space, and speak of operations on the manifold while they are actually defined on $\mathbf{R}^n$ through the bijective correspondence). This gives us a nice way of introducing coordinates on an arbitrary space (within reason of course), for example a sphere, a curved space-time (as in relativity) or even abstract things (e.g. I was told manifolds are also the most natural way to describe a physical experiment, the space existing of all outcomes, or something like that).

 

[Phred101.2]

Manifolds are (just) a 'surface' that is described mathematically. Topology math uses n-dimensions and hyberbolic and imaginary spaces, but they are just projections of what you do when you integrate a curve, say, or project any lower-dimensional 'function' onto a higher dimensional space.. somethink like that.

 

[HallsofIvy]

A "manifold" is a space that is "locally" the same as Euclidean n-space. Specifically, there must be a collection of "pairs" $(U_a, f_a)$ so that the $U_a$ are open sets that "cover" the space (every point in the manifold is in at least one) and every $f_a$ is a homomorphism (cotinuous invertible function whose inverse is continuous) from $U_a$ to a subset of Rⁿ. That means that given any point p in the manifold, we can identify some neighborhood of it with a neighborhood of Rⁿ and so assign "coordinates" to it.

 

[cesiumfrog]

An example two dimensional manifold is the surface of a doughnut, as contrasted with the normal (flat, infinite) two dimensional Euclidean space that you're familiar with. Every manifold can be described as a "(n-dimensional) surface" embedded (but curved) in some higher dimensional Euclidean space. The things that distinguish n-dimensional manifolds from normal n-dimensional space are "geometry" (once a metric is determined you may find that parallel lines do not remain equally separated, and the ratio of circumference to diameter may only be Pi for infinitesimally small circles) and "topology" (if you go in a straight line you may still end up back where you started).

 

[theperthvan]

What about watching a movie? It's kind of like squashing 3D into a flat screen. Like a backwards manifold. Is there some object that describes that?

 

[pmb_phy]

Request: I posted a link to my web page that defines the term "manifold". I would like everyone's opinion who is versed in differential geometry and tensor analysis to give me their opinion as far as the accuracy of the definition and the quality of the description. Perhaps I left something out? I noticed here that someone mentioned a bijection, which of course is true. I mention the math of the bijection but didn't see the need to use the term. I think that was a mistake since its a good example of where the terminology finds ues.

Thanks

Pete

 

[trinitron]

"What about watching a movie? It's kind of like squashing 3D into a flat screen. Like a backwards manifold. Is there some object that describes that?"

Projection.

 

[Phred101.2]

Bijection, surjection, and so on all describe mapping or projection. All math ultimately projects "back" to numbers and counting or measuring, recording changes like increase (in population, say) and decrease. It's all projection from there on up. The idea of a surface is a projection of some line or curve, and this holds in (almost) any space... (don't quote me on that last phrase).

 

[zankaon]

manifold i.e. continuum

A poor man's simple definition, would just be to consider manifold as a continuum; and continuum is just the inbetweenness quality in the matching of sets' respective elements. For example, pseudo-Riemannian spacetime (C_R) or continuum or manifold, would all be ok. More specifically, for consideration of 2 neighboring patches, one might consider if they are topologically equivalent; that is share the same continuum i.e. have the same inbetweenness quality. The minimum number of elements of a set, in order to consider inbetweenness would be 3! any higher finite or transfinite would then also be acceptable, in math. Another example of manifolds, would be Minkowski tangent spacetime, tangent to C_R manifold. The apple depiction on MTW Gravitation book is inaccurate, in that Minkowski spacetime is just an idealized tangent manifold. That is, there is always geodesic deviation (convergence), even for 2 objects in freefall towards the CM of earth-moon; hence always curvature, and thus always mass-energy. see R. Penrose Road to Reality. very readable with emphasis on manifolds.