One of my professors wrote the following open question on the blackboard:
If $M$ is a compact, connected smooth $4$-manifold such that $\pi_1(M) = 0$, $\pi_2(M) = 0$ (first two homotopy groups are trivial), does it follow that $M$ is diffeomorphic to the $4$-sphere?
and warned us, that if we managed to solve it, we would get an instant Ph.D. -- so, keen on getting a Ph.D. before my bachelor degree, I went to work immediately! ;-)
My first thought was the following: If one could endow $M$ with a Riemannian metric giving a Riemannian manifold with constant sectional curvature $1$, then by compactness $M$ would be a complete, connected, simply connected manifold of curvature $1$, which would imply the statement.
Now I obviously didn't get much further than this (my dreams were shattered!).
Anyways, this leads to the question:
"When is it possible to endow a smooth manifold with a metric which has some desired properties (i.e. constant curvature or bounded curvature)?"
Has there been much work on this? Are there any good books/papers I could take a look at (just to get some impression of how the experts approach this problem)?
I was also wondering whether the above is actually an approach to the problem taken by people working in the field? Or may it be completely hopeless to try and gain any control of the metric globally?
Well, as always I thank in advance for any comments, answers etc.
Best regards, S.L.