Examples of global properties that don't arise from local knowledge

Question

Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by $$ 2\pi \, \chi(M) = \int_M K_p \; dA $$ where $dA$ is the Area element associated to $g$ and $K_p$ is the scalar curvature at a point $p \in M$ (Gauss-Bonnet).

On the other hand there are global properties that don't arise in this fashion, i.e. that don't arise from local properties alone. For instance uniform continuity cannot be deduced from continuity at each point of the domain of some function.

I am interested in examples from differential geometry, i.e examples where global properties do not arise from local properties alone. Would orientation serve as a good example or are better or more insightful ones ?

Answer 1

Orientation is a good example that you mentioned. A somewhat related example is the difference between a closed and an exact differential $k$-form. The former is local but the latter is global; for example the 1-form $d\theta$ on the unit circle is easily seen to be closed locally but to see that it is not exact one needs to take an integral. Stokes' theorem for differential forms is a generalisation of this example.

An interesting perspective on the local/global dichotomy is provided by Abraham Robinson's framework. Here it turns out that certain properties that appear to be global in nature from the viewpoint of the reals $\mathbb{R}$ turn out to behave like local objects when viewed from the perspective of the hyperreals ${}^{\ast}\mathbb{R}$.

An example is uniform continuity that you mentioned. A real function $f$ is uniformly continuous on $\mathbb{R}$ if and only if its natural extension is S-continuous at every point of ${}^{\ast}\mathbb{R}$. Here S-continuity is a local property but it serves to characterize a global property of a real function.

Another good example in this context is the Dirac delta "function". This is generally interpreted in terms of Schwartzian distributions, and of course the physicists' description of it as a function with local values (infinity at the origin, zero elsewhere, total integral $1$) is naive. It turns out that in Robinson's framework there does exist an (internal) function with local values such that, when integrated against a continuous function $f$, will produce the value of the function at $0$. This is another example of a property generally thought to be global which turns out to be local in an extended framework.

Answer 2

A simple example is boundedness: you can't deduce that a function is bounded based only on its local behavior. Said another way, the presheaf of bounded functions is not in general a sheaf.

Answer 3

Quite a lot of topology can be done locally. The paper by Gelfand and Tsygan, Localization of Topological Invariants, has a general construction of what can come from one method of integration of local quantities. Also anything from rational homotopy, which can be built (a la Sullivan) from differential forms. Index of elliptic operators is another and very famous example.

Answer 4

If we intend as local property a property that can be deduced from a neighborhood of a given point, than it seems to me that in your example you are using (implicitly) a global property, i.e. the fact that the manifold is compact (as far as I know this property cannot be deduced locally).

In this sense as a classical example of two manifolds locally diffeomorphic but with different global properties you can think at a square in $\mathbb{R}^2$ with the opposite sides identified (a flat torus) that locally appears as a flat $\mathbb{R}^2$ space, but has loops that are not null-hopotopic.

Answer 5

Simple connectedness is a very simple (sorry for the bad pun) example.

Answer 6

You can define Euler class for any oriented bundle. Chern-Weil theory gives construction of Euler class from chosen $O_n$ connection (via Pfaffian). But if you do not require a connection to be orthogonal, then you do not have such description.

Moreover, there are bundles with non zero Euler class, which admits flat connection.