I am an undergraduate currently taking a first course in smooth manifolds. I feel that I understand the material intuitively. But, I'm having trouble turning my intuition into proofs. I was hoping that I could get answers to the two following questions:
- What are the essential basic tactics for attacking smooth manifold and differential topology problems?
- What are the best things to do to improve my technique and ability to solve problems?
Let me give an example of what I mean. I just asked another question about showing a particular map is a submersion. I have an intuitive understanding of what is going on. In the context of the problem, I can see what is happening in the case $\pi: SO_3 \to S^2$. The idea is that if I look at a point on a sphere, and I look at the directions coming off of that point, there is a rotation of the sphere that moves it in that direction. For instance, if I'm looking at a point on the equator, there is a rotation that moves the point northward, and a rotation that moves it westward, and that's enough to show I can move the point in any direction I like.
But, turning that into a proof is a little bit of a daunting task. So daunting, in fact, I was compelled to ask about it. When I asked my professor about this problem, he drew me a picture and explained the intuition. I think intuition is important, but what I'm having trouble with it turning my intuition into a proof. The reason I'm having trouble with that is I don't exactly know what the essential proof techniques are. So, I firstly want to know what these proof techniques are, and secondly want to know what I can do to be acquainted with their use.
I feel I should clarify what I'm saying somewhat. When I say tools here, I mean something along the lines of "use Sard's theorem" or "partitions of unity." For example, I would say that using the regular level set theorem, instead of constructing charts, to show that some set is a manifold is a tool. I understand that showing something is s submersion requires showing its differential is surjective, that being the definition of the term. What I mean is, what are good ways to exploit the structure I have? What are the proof techniques unique to this structure?