Question

How do we know manifolds are the "correct" objects to study?

Answer 1

I mean your question really applies to everything in math. Its not so much a question of whether things precisely define the world around us, its how well those definitions agree with our intuition. For instance, you can't write a proof that real numbers are worth studying or actually mean anything, but we understand them intuitively and our definitions agree with that inuition pretty well, so they end up being relatively useful. For manifolds, they formalize our inuition of a locally nice looking space, so if you care about that then theres reason to study them.

Answer 2

As others have pointed out tbere are structures that still preserve some parts, but lose others, so I want to mention another such family of spaces: geodesic metric spaces.

A geodesic metric space is simply a metric space where any 2 points can be connected with a distance minimizing path/segment.

In these you can do a lot of stuff you would do with curvature in Riemannian manifolds (manifolds with a (Riemannianl metric), usually by comparing a triangle in you manifold to one in a space of constant curvature.

Not too sure how much it simplifies any proofs, but it generalizes curvature to stuff like graphs or stuff that would be infinite dimensional if you tried looking at them like manifolds. Notably a lot of the constructions you can do on negative/non-positive curvature manifolds can be done mostly on CAT(0) spaces (e.g. visual boundary).

Answer 3

One important thng to remember is that smooth manifolds were born as a means to talk about regular surfaces of euclidean space without needing to worry about an ambient space.  There are several reasons why this is useful, for one physical motivation, consider the case of General Relativity. According to it, the universe is a 4-dimensional manifold which need not be euclidian, so physically, what would be an ambient space for the "space" itself? This is just one of the many reasons one would to want to work with an intrinsic definition of "regular surface".

Of course, you might wonder if indeed manifolds are really just regular  in desguise. The answer is yes! In fact, we have embedding theorems that say that every manifold is diffemorpic to some regular surface of some Rn. This, at least to me, shows that the definition is good for what it intented to do: every regular surface is a manifold, but also, every manifold is a regular surface.

Of course, as with any other defintion, we can remove some requirements and end up with more general objects, which often times are also very interesting on their own. For instance, if you remove the smoothness of the change of coordinates, you end up with a topological manifold, which have been studied extensively.

Answer 4

They work. And nobody came up with anything that works better. Yet.

Answer 5

There certainly ARE other types of similar objects that can be studied. Many statements about manifolds translate or copy over to the worlds of simplicial complexes, CW complexes, algebraic varieties, etc.

Most statements about Lie groups can be reformulated for linear algebraic groups.

This idea of there being multiple analogous constructions is not uncommon in math, and when it matters, you choose the one that does what you want. Manifolds are a good class because they are relatively flexible but not too open. They connect your space to an object we understand well- R^n- via nice maps, and there are classification theorems for manifolds.

Answer 6

Other objects are frequently used. For example, manifolds with corners, or analytic varieties, or orbifolds. All of these have different uses, it just so happens that manifolds are probably closer to things you have seen before, so they are defined earlier in your education. Even if we did find a class of objects that agrees with manifolds for low dimensions and is easy to classify in high dimensions we still wouldn't stop studying manifolds. They are too interesting.

Answer 7

>This is hardly surprising---manifolds are central to many mathematical and physical phenomena.

You have answered yourself.

>How do we know there isn't some slight modification to the definition of a manifold

There are. Hibbert space, Sobolev spaces, Hausdorff space, Zariski topology...etc all slightly different.

There are more exotic objects out there. The one mentioned most often is obviously of the most interest.

Answer 8

Because manifolds capture the essence of a lot of spaces we are interested in. A theory is justified by its examples and applications.

Answer 9

The Whitney Embedding theorem.

Answer 10

We don't.

But the way maths works is that such candidates are proposed and studied and if shown to be more interesting etc. then the hive mind will gravitate towards it. Main objects of study do change over time.