ExampleConstruction: Literature-sourcedVerification: Literature-sourcedjudge pass (gpt-5.6-terra)audited 2026-09-04

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A standard torus height function has four critical points

Example

On the torus $T^{2} = R^{2} / Z^{2}$ , the smooth function

$f ([x], [y]) = \cos (2 π x) + \cos (2 π y)$

has exactly four critical points: one minimum, two saddles, and one maximum. It is Morse but not excellent.

Facts & Assumptions

Given: The torus function $f ([x], [y]) = \cos (2 π x) + \cos (2 π y)$ .

[F1]

Morse, excellent, nondegenerate, and index are defined on the A page (Morse functions and excellent Morse functions, Nondegenerate critical points, nullity, index, and coindex).

Verification

technique · direct computation

1.1

The partial derivatives are $\partial f / \partial x = - 2 π \sin (2 π x)$ and $\partial f / \partial y = - 2 π \sin (2 π y)$ , so a point is critical exactly when $x, y \in {0, \frac{1}{2}}$ mod $Z$ . Hence there are exactly four critical points.

givenalgebra

2.1

The Hessian matrix is $diag (- 4 π^{2} \cos (2 π x), - 4 π^{2} \cos (2 π y))$ . At $(0, 0)$ it is negative definite, at $(\frac{1}{2}, \frac{1}{2})$ it is positive definite, and at $(0, \frac{1}{2})$ and $(\frac{1}{2}, 0)$ it has one positive and one negative eigenvalue. Therefore the four critical points have indices $2, 0, 1, 1$ respectively, and all are nondegenerate.

F1step 1.1algebra

3.1

Hence $f$ is Morse by [F1]. Its critical values are $2, - 2, 0, 0$ , so the two saddles share the value $0$ . Therefore $f$ is not excellent.

F1step 2.1algebra∎

Depends on

Used by

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Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)