The only closed manifolds which allow a function with two (maybe degenerate) critical points are spheres. In dimension 2 it is quite easy to prove, but what is about higher dimensions?
2026-03-25 14:20:07.1774448407
Manifold allowing function with two critical points is sphere
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Topologically the answer is yes, but in the smooth category the answer is no. Milnor provided the first examples of topological spheres which are not diffeomorphic to the standard sphere. His argument that they are toplogical spheres uses the fact that there are Morse functions with two critical points. This wikipedia page should get you started .