Global Coordinates in Differential Geometry?

Question

In trying to learn a bit about differential geometry I have hit a puzzler. Most texts emphasize that one coordinate system will not suffice in general, but the reasoning is never given. After all, if a collection of open sets (the domains of the various coordinate mappings) cover the manifold and each is diffeomorphic with an open set of $R^n$, then why can't the local diffeomorphisms be "continued" so as to constitute a mapping of the whole manifold diffeomorphically to $R^n$? (In much the same way that a function analytic on an open set in the complex plane can be analytically extended to the entire plane.)

This would allow one to establish a common origin for the coordinates, thereby placing all the onus of "nonlinearity" on the (global) coordinate mapping.

I'm sure I am missing something, but could someone please tell me precisely what?

Thanks.

Answer 1

The sphere can't be covered by one cooridnate patch. Suppose there is an open set in $\mathbb{R}^2$ diffeomorphic to the whole sphere, note sphere is compact, in particular closed and bounded $\mathbb{R}^3$. By continuity of inverse of the parametrization, the sphere should be open. Hence we know there is an proper subset in $\mathbb{R}^3$ clopen, which contradicts to connectedness of $\mathbb{R}^3$.

Answer 2

If you can find a global coordinate system then the coordinate vector fields are nontrivial on the manifold. However, there exist manifolds where this is not possible. Even something as simple as a sphere. The local nature of coordinates is not an oversight of the theory, it is an essential feature.

Answer 3

First, a function analytic on an open set can not necessarily be analytically continued to all of $\mathbb C$. Take $1\over{z-1}$ on the open unit disc, for an example. Or the $\Gamma$ function, or the Riemann zeta function, or any other meromorphic function. And even better, try exponentiating any of those and you'll be analytic on an open set but have essential singularities in any attempt to extend to the whole plane.

To continue the complex analysis examples, you can see what goes wrong in an attempt to construct a global chart from local ones by considering $\log z$. Pick any branch, and now try to get a single well-defined global version by approaching the branch cut from opposite sides. You will find the imaginary parts disagree by $2\pi i$ no matter what you do.

This is the crux of the issue in a manifold: there is nothing a priori that assures you won't get complete nonsense out of patching things together locally into a global picture. The definition is explicitly local in statement, and the pursuit of theorems that can be stated globally is a generally non-trivial endeavor.

As others have noted, there are standard topological reasons why a given manifold can't be diffeomorphic to $\mathbb R^n$. Connectedness, compactness, homology groups, (higher) fundamental groups, etc. are all preserved by homeomorphisms, and thus diffeomorphisms. For example: $S^n$ is compact, $\mathbb R^n$ is not. Thus they are not diffeomorphic.