What is global differential geometry?

What is the difference between local and global differential geometry? I cannot find their (exact) definitions. There are some other terms in geometry like "rigid" (e.g. that structure is more rigid that the other one) which it seems they don't care to define. I think I vaguely know what they mean but want to be sure.

Solution 1:

Let me expand Thomas' good answer by saying that "local" differential geometry is the study of properties of a geometric structure that (at each point) depend only on an arbitrary neighborhood of a given point, or more precisely the germ of the structure at that point. A standard example is the curvature of a Riemannian metric: Its value at a point only depends on local data (in fact, it only depends on two derivatives of the metric at the point), and so it doesn't change if we, say, remove all of the manifold except an arbitrary neighborhood of the point.

A general example: If one has a notion of a "flat" instance of a geometric structure (typically a homogeneous structure), one can ask whether a given structure is locally flat, i.e., locally diffeomorphic (around each point) to the flat structure, and often the existence of such a local diffeomorphism is obstructed exactly by some curvature quantity. For example, we say a Riemannian structure $(M, g)$ is locally flat if at each point it admits a local diffeomorphism of $(M, g)$ with the Euclidean metric $(\mathbb{R}^n, \bar{g})$, and $(M, g)$ is locally flat iff its Riemannian curvature $R$ vanishes. Local flatness doesn't tell us, however, whether a manifold is, e.g., (globally) diffeomorphic to $(\mathbb{R}^n, \bar{g})$.

Global properties, however, deal with the entire manifold (or at least properties of some region larger than an arbitrarily small neighborhood)---properties that can change when you alter a part of the manifold.

There are some geometries that have no local invariants, and for these geometries only global questions are interesting. For example, given a contact structure on a $3$-manifold $M$---that is, a $2$-plane distribution $\bf H$ such that $[{\bf H}, {\bf H}] = TM$---we can always find local coordinates $(x, y, z)$ around any point of $M$ in which the contact structure is given by ${\bf H} = \ker (dy - x \,dz)$. So, there is no equivalent of the Riemannian curvature $R$ for this geometry, and (unless one adds additional structure to $\bf H$) the only interesting questions about this geometry are global.

Some important theorems relate local and global geometry, and realize global properties as integrals of local ones: A typical example is the Gauss-Bonnet Theorem.

Solution 2:

Global differential geometry deals with the geometry of whole manifolds and makes statements about, e.g., the diameter, the minimal number of closed geodesics or whether a manifold has be (non-)compact by analyzing geometric quantities like the curvature. Opposed to this is the local study of balls, whether they are, say, geodesically convex. I never heard the term 'local differential geometry', I have to admit. I also don't think there is a strict definition of the term, but may be wrong.