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

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The height function on the sphere is Morse and excellent

Example

For $n \geq 1$ , the height function

$h : S^{n} \to R, h (x_{1}, \dots, x_{n + 1}) = x_{n + 1}$

has exactly two critical points, the south and north poles. Their indices are $0$ and $n$ , so $h$ is Morse and excellent.

Facts & Assumptions

Given: The height function $h (x) = x_{n + 1}$ on the unit sphere $S^{n}$ .

[F1]

Critical points, nondegeneracy, index, and Morse/excellent functions have the meanings fixed on the A page (Critical points and critical values of a smooth function, Nondegenerate critical points, nullity, index, and coindex, Morse functions and excellent Morse functions).

Verification

technique · direct local model

1.1

If $x \in S^{n}$ is not a pole, let $v : = e_{n + 1} - x_{n + 1} x$ . Then $x \cdot v = 0$ , so $v \in T_{x} S^{n}$ , and $d h_{x} (v) = v_{n + 1} = 1 - x_{n + 1}^{2} \neq 0$ . Hence only the poles can be critical.

F1givenalgebra

2.1

Near the north pole, write the upper hemisphere as $u \mapsto (u, \sqrt{1 - ∥ u ∥^{2}})$ . Then $h (u) = \sqrt{1 - ∥ u ∥^{2}} = 1 - \frac{1}{2} ∥ u ∥^{2} + O (∥ u ∥^{4})$ , so the Hessian at $u = 0$ is $- I_{n}$ and the north pole has index $n$ .

step 1.1algebra

2.2

Near the south pole, use $u \mapsto (u, - \sqrt{1 - ∥ u ∥^{2}})$ . Then $h (u) = - \sqrt{1 - ∥ u ∥^{2}} = - 1 + \frac{1}{2} ∥ u ∥^{2} + O (∥ u ∥^{4})$ , so the Hessian at $u = 0$ is $I_{n}$ and the south pole has index $0$ .

step 1.1algebra

3.1

Steps 1.1-2.2 give exactly two nondegenerate critical points with distinct critical values $- 1$ and $1$ . Therefore $h$ is Morse and excellent by [F1].

F1step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michele Audin and Mihai Damian, Morse Theory and Floer Homology (standard reference, not scraped)
Liviu I. Nicolaescu, An Invitation to Morse Theory, 2nd ed. (standard reference, not scraped)