A Morse function is a smooth real function on a smooth manifold without boundary whose critical points are all nondegenerate. This means that the Hessian in a coordinate chart is a nonsingular matrix at every critical point (Fuchs n.d., pp. 36, 38). Distinct values at distinct critical points are an additional convention that can be achieved without changing the critical point count.
Morse Function
See also
Critical Point, Kontsevich Integral, Morse Knot, Morse Number, Morse Theory, Stable Morse NumberPortions of this entry contributed by Sergei Duzhin
Portions of this entry contributed by Sergei Chmutov
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References
Fuchs, D. Notes on Manifolds. Undated lecture notes, pp. 36, 38, 42. https://www.math.ucdavis.edu/static/research/Fuchs_Profile-MAT239_Notes_on_Manifolds.pdf.Referenced on Wolfram|Alpha
Morse FunctionCite this as:
Weisstein, Eric W., with contributions by Sergei Duzhin and Sergei Chmutov. "Morse Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MorseFunction.html