Does Differential Topology or Differential Geometry play a larger role in Chaos Theory?

Question

I'm an undergraduate on somewhat of a time constraint in school. I have room in my remaining schedule for a semester of either Differential Geometry or Differential Topology.

I understand the topological side deals with primarily global aspects of a manifold, whereas the geometrical side associates with interesting local structures.

To be concise- I'm interested in chaotic dynamical systems and P.D.E., so which subject would be more worthwhile, i.e. which subject's vocabulary/method would equip for further reading in the topics I am interested in? Or am i naive in forcing myself to choose between the two topics, as similar as they are?

Thank you!

Answer 1

I think differential topology is better to know for dynamical systems. For example, results like the Poincaré-Hopf index theorem are used.

On the other hand, I would say PDEs goes well with differential geometry. But I think knowing PDEs helps a differential geometer more than knowing differential geometry will help someone with PDES-- many questions about geometry can be phrased in terms of PDEs.

Answer 2

I think it makes sense to learn topology really well. Because in differential geometry you still use topology notions like immersion, transversality all the time. In differential topology even though people use a good amount of differential geometry, you usually do not need to compute the Riemann curvature tensor or solve the Jacobi equation explicitly. The curvature rules in differential geometry, but played a limited role in differential topology until one encounter slightly more advanced topics like flat connections and holonomy group.

Also for a one semester class, it is likely that you will learn differential topology(Hirsh's book) well enough to be able to read more advanced texts, while for differential geometry it is difficult to cover that much material. Most graduate students I know take at least a year to learn Riemannian geometry - Comparison geometry really well. So it is up to you to decide. Also, people's learn style differ a lot. It is "well known" that S.T.Yau finished 7 graduate level courses in one semester and audited 7 other graduate level math courses at 21 or 22 in Berkeley. Maybe you can audit both classes and see which one you like.

Answer 3

I took both. I feel differential geometry is more tangible and computational, while differential topology is more conceptual.