Skip to main content
2 of 3
added 186 characters in body
C.F.G
  • 8.6k
  • 2
  • 22
  • 57

Morse theory says that every Morse function $f$ (that is all critical points were non-degenerate and distinct critical points take distinct critical values.) $\#\min+\#\max-\#\mathrm{saddle}=\chi(M)$. So, in the case of torus (that its Euler char is $0$), the functions must have $\#\mathrm{saddle}=\#\max+2$. By the fact that all continuous functions on a compact domain attaint at least a max and a min therefore we should have at least a max point then at least 3 saddle point for torus. in the case of sphere is similar. $\#\min+\#\max-\#\mathrm{saddle}=\chi(\Bbb S^2)=2$ so having two global minima we must have $\#\max=\#\mathrm{saddle}\neq 0$.

Note that these are non-degenerate critical points (that means Hessian is nonsingular at that points) and there is a function on torus with 3 critical points i.e. a min, a max and a degenerated saddle point.

C.F.G
  • 8.6k
  • 2
  • 22
  • 57