How do we know that we can always make coordinate charts everywhere on a manifold?
If we have two sets of coordinate systems, how do we know if they represent the same manifold or different manifolds?
An even bigger question: if we have two manifolds, what information about each manifold do we need in order to know whether they're different manifolds, or actually the same manifold?
Further Questions
If an ant walks in a straight line around the surface of a manifold, what path will it take? Will it return to its starting point? If so, after how long? Are there shorter paths? In general, "paths on manifolds" is full of interesting questions to ask.
A "3-torus" can't exist in ℝ3 without self-intersecting, but it can in ℝ4. If you have an n-manifold, what's the smallest ℝm it can exist in without self-intersecting?
Each coordinate chart of an n-manifold is a coordinate system with n real numbers. What if instead of real numbers, we made each coordinate system out of n complex numbers?
Our universe sure seems like a 3-manifold. Which 3-manifold do we live in?