I am trying to show that a smooth function $f:S^1\times S^1\to \mathbb R$ must have more than two critical points. Since $f$ attains maximum and minimum, it must have at least two critical points. How would one show that they can't be two?
If one considers the gradient vector field $\nabla f$ then by the Poincare Hopf index theorem its index is equal to the Euler characteristic of the torus, i.e. it is $0$. Therefore $\nabla f$ has an even number of zeros.
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