Differential Topology: Essential Concepts Explained

[psholtz]

I have what's certainly a totally "newbie" question, but it's something I've been wondering about..

Suppose we have a simple boundary value problem from electrostatics. For instance, suppose we have a conducting sphere held at some potential, $$\phi = \phi_0$$. Because the sphere is conducting, and b/c we are dealing w/ the static case, take the entire surface of the sphere to be at the same potential ($$\phi_0$$ in this case).

We now wish to solve Laplace's Equation, $$\nabla^2 \phi = 0$$

Note first that the surface of the sphere divides $$R^3$$ into two "topologically distinct" spaces: first there is the topologically "closed" region, which is the interior of the sphere, and second there is the toplogically "open" region, which is everything external to the sphere.

Note also the solutions to Laplace's equation (which in both cases are trivial) take two distinct forms in each topological region: in the interior (i.e., closed) region, the solution is to take the potential to be a constant, specifically $$\phi = \phi_0$$, which in the exterior (i.e., open) region, the solution is to take the potential as a 1/r function, specifically $$\phi = Q/r$$ for a constant Q.

My point in going through this relatively simple example is simply this: here we have the same differential equation (Laplace's equation), subject to the same boundary condition (potential constant on the surface of a sphere), and yet we derive two distinct solutions to the DE depending on whether we are solving the system in a topologically closed region, versus a topologically open region.

Is this the essence of what differential topology, or differential geometry, concern themselves w/?

 

[Tac-Tics]

It doesn't matter if you consider the sphere itself to be part of the interior or the exterior (or you ignore it all together). So, the interior isn't necessarily closed and the exterior isn't necessarily open. The reason for the behaviors of the potential don't depend on the openness or closedness of the space.

Differential geometry is something pretty different. It is the study of differentiable manifolds. Manifolds are topological spaces which are locally homeomorphic to Euclidean space. The best example is the surface of the Earth which "looks" 2 dimensional, even though it's not planar.

If you have a well-behaved manifold, you can do calculus on it. We call these differentiable manifolds. With these manifolds, we can study derivatives. On a sphere, we can pictorial represent derivatives as vectors which are tangent to it.

If you have a well-behaved differential manifold, you can also take the dot product of vectors. We call these Riemannian manifolds. Riemnnian manifolds have a way to measure lengths of paths on the manifold, as well as a way to define "straight lines" (relative to the manifold) called geodesics.

If we loosen up on what "dot product" means, we end up with pseudo Riemannian manifolds, which are the basis for general relativity in physics.

 

[wofsy]

Your question is really good. The topology of a domain has a strong effect upon solution of PDE's such as Laplace's equation. And I think it is true that early ideas in topology came in part from attempts to solve Laplace's equation, e.g. the idea of Riemann surfaces.

In modern times the study of PDE's on manifolds has blossomed and has led to many profound results such as the solution of the Poincare conjecture in dimension 3. Since the Laplacian requires a metric in order to be deinfed - i.e. its definition requires a notion of shape - studies of solution of the Laplacian and the Heat Equation require differential geometry and lead to many geometric theorems. One can for instance prove the Hodge Theorem and the Gauss Bonnet Theorem by studying solutions of the heat equation on manifolds.

A famous illustrative paper is Marc kac's "Can You hear the Shapre of a Drum?" which studies whether the spectrum of the Laplacian on a two dimensional bounded domain determines the domain's topology.

Differential topology on the other hand does not require a metric - and thus is not a natural arena for studying the Laplace equation and its relatives. On the other hand one can study the global behavior of solutions of ODE's and arrive at theorems that do not depend on the particular equation. This area of mathematics was initiated by Poincare at the turn of the 20'th century and has many offshoots in modern times e.g. the study of mathematical chaos
and the proof of the Poincare conjecture in dimensions 5 and above.

The connection between differential equations and topology and geometry is huge. It is an exaggeration but still apt to say that most mathematics was developed to solve differential equations.

 

[psholtz]

It doesn't matter if you consider the sphere itself to be part of the interior or the exterior (or you ignore it all together). So, the interior isn't necessarily closed and the exterior isn't necessarily open. The reason for the behaviors of the potential don't depend on the openness or closedness of the space.

Perhaps, but the interior of the sphere is bounded, and the space exterior to the sphere is unbounded, correct?

Differential geometry is something pretty different. It is the study of differentiable manifolds. Manifolds are topological spaces which are locally homeomorphic to Euclidean space. The best example is the surface of the Earth which "looks" 2 dimensional, even though it's not planar.

So a differential manifold is a vector space of at most (n-1) dimensions, embeded in a vector space of n (or more) dimensions?

 

[Tac-Tics]

Perhaps, but the interior of the sphere is bounded, and the space exterior to the sphere is unbounded, correct?

That much is true. What effect that has on possible scalar functions, I don't know.

So a differential manifold is a vector space of at most (n-1) dimensions, embeded in a vector space of n (or more) dimensions?

That's the intuition. But there is a little more to the actual definition.