The critical point number 
of a closed smooth manifold 
is the minimum
number of critical points among all smooth real
functions on 
. Degenerate critical
points are allowed, in contrast to the Morse number.
For a positive-dimensional sphere
the minimum is 2, attained by a height function.
The rational cup length
gives 
. The extra 1 corresponds to the
unit-inclusive convention for cup length. This invariant
is different from a critical number in elementary calculus, which denotes the location
of a critical point rather than a minimum
count.
Critical Point Number
See also
Arnold Conjecture, Critical Point, Cup Length, Morse NumberExplore with Wolfram|Alpha
References
Cannas da Silva, A. Lectures on Symplectic Geometry. Berlin: Springer-Verlag, 2001. https://people.math.ethz.ch/~acannas/Papers/lsg.pdf.Cornea, O.; Lupton, G.; Oprea, J.; and Tanré, D. Lusternik-Schnirelmann Category. Providence, RI: Amer. Math. Soc., 2003. https://doi.org/10.1090/surv/103.Cite this as:
Weisstein, Eric W. "Critical Point Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CriticalPointNumber.html