What is a Manifold?

Now we always encounter definition of a manifold from a mathematical point of view where it is a topological space along with a family of open sets that covers it and the same old symphony. My question is from your own expertise and from what you have learned and taught to day, how can we get a deeper understanding to what's a Manifold? Maybe from a more physics-point of view?

Solution 1:

G. Bergeron's answer is good and gives sound advice. However, as you grapple with the modern manifold concept, it may help you to know some of the history behind the idea, so that you can see it some of the more seemingly abstract parts of it didn't come out of nowhere and emphasise G. Bergeron's quote

The rigorous mathematical definition is not there to annoy or obfuscate.

The modern notion was fully finished in the 1940s by Hassler Whitney. Before that, there were two notions of manifold floating around, which mathematicians suspected ultimately described the same notion at some level, but it fell to Hassler Whitney to prove it. Firstly, there was a more obvious, less abstract one of a smoothly constrained subset of $\mathbb{R}^N$, thus one thought of a geometric object as being embedded in a higher dimensional Euclidean space: a 2 sphere defined by the constraint $x^2+y^2+z^2=1$, for example. So, instead of a collection of patches (charts), we have a kind of global superchart that is a constraint like the one I've just cited. Then there was the modern definition: I love user garyp's pithy intuitive summary:

I use this: a manifold is a space that is locally Euclidean, but globally might be complicated, e.g. a torus or sphere, or etc.

This really is just about all there is too it: there really is only one background detail you need to become well acquainted with and that is the details about how the transition maps glue the patches i.e. the open sets + coordinate maps that define the local morphism - (homeo-, diffeo-, smooth diffeo- analytic diffeo-morphism- depending on what kind - topological, differential, smooth, analytic ....- of manifold we're dealing with) together. Whilst these may seem dense and abstract they have a simple and easily stated purpose: to make sure that the "gluing" is "seamless" i.e. that assertions you prove or true statements you make with one co-ordinate map are true with any other co-ordinate map that applies to an overlapping region and contrariwise. I personally also like to think of the overlapping mechanism (i.e. why a manifold is a collection of overlapping patches rather than a partitioned set) as an abstraction of a concrete - and Copernican - physical notion: no seafarer expects to come against a wall in the middle of the ocean just because his or her chart gives out. The chart edges are an artifact of our description, not the actual mathematical object. So there should always be an equally good description with no wall at the place in question. Also, there is the mathematical practicality that you want every point of the manifold to be inside and not on the edge of some set, particularly if you're doing calculus or differential geometry. You want a one-size-fits all, seamless, definition: you don't want to have to be dealing with onesided limits at edges of a partition.

I suspect that the modern notion had at least some boost from physics (classical at least): once GTR came along with its fundamental notions of locality, you really do need to switch to a local definition. In principle one could imagine GR done on some object embedded in a higher dimensional Euclidean space, but think of all the redundant, non-physical information you'd have to grapple with. The embedding defines the manifold as a relationship between it and its complement in the higher dimensional space and the latter, as far as we know, has no physical interpretation. GR already has quite enough redundancy to grapple with and an embedded-in-Euclidean space theory would probably be utterly intractable. It would, at the very least, be a great deal more complicated.

The two historical notions of manifold were united by the Whitney Embedding Theorem, achieved through the famous Whitney Trick (summarized in the Wiki article). The second, more interesting from a GR standpoint, is the Nash Embedding Theorem. Both these show that we can embed any manifold defined by the modern notion in a higher dimensional Euclidean space (or Minkowski space, in the case of a pseudo-Riemannian manifold such as described by GTR). The Nash theorem shows we can do this isometrically. The practicalities of these powerful sounding theorems though are formidable and thorny: you may need to use a $2 m$ dimensional Euclidean / Minkowski space to Whitney-embed an $m$ dimensional manifold in. The Nash theorem is "even worse": an $m$ dimensional manifold may need as high as $m(3m+11)/2$ Euclidean dimensions if it is compact. If it is not compact, you really draw the short straw as you may need to go to $m(m+1)(3m+11)/2$ Euclidean dimensions. These are saturable bounds: you can be lucky in special cases and need fewer.

Solution 2:

A manifold is before all a mathematical object. As such, any deeper understanding of a manifold per se will be gained from a rigorous mathematical study of the object. From a physics point of view, manifolds can be used to model substantially different realities: A phase space can be a manifold, the universe can be a manifold, etc. and often the manifolds will come with considerable additional structure. Hence, physics is not the place to gain an understanding of a manifold by itself.

The rigorous mathematical definition is not there to annoy or obfuscate, it is to ensure that the object is well defined and have a structure that is also well defined, from which we can derive theorems and intuitions. The connection with physics happens when a particular mathematical abstract object reflects the structure of the physical concepts under scrutiny. From there, we construct model using these mathematical objects to represent a physical concept in the mathematical model.

That being said, I would say the closest to an answer I can give to your question is that manifolds are the mathematical objects representing the intuitive notion of geometrical spaces. A topological space is too weak in that respect and goes a bit beyond common sense intuition, while a smooth manifold is the correct object when talking about the idea of a space which you would want to parametrize with coordinates (at least locally), but even then, a rigorous study shows you that these ideas cannot be worked with too naively.