LemmaStatement: Literature-sourcedProof: Literature-sourcedaudited 2026-09-04

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A nonzero second derivative splits off a signed square with a smooth parameter

Statement

Let $m \geq 0$ , let $U \subseteq R \times R^{m}$ be open around $(0, 0)$ , and let $F : U \to R$ be smooth with

$F (0, 0) = 0, \frac{\partial F}{\partial u} (0, 0) = 0, \frac{\partial^{2} F}{\partial u^{2}} (0, 0) \neq 0.$

Then, after shrinking $U$ , there are a sign $ε \in {\pm 1}$ , a smooth function $H$ of the parameter variable, and a smooth local coordinate change $(u, y) \mapsto (u^{'}, y)$ fixing $(0, 0)$ such that

$F (u, y) = H (y) + ε (u^{'})^{2} .$

Facts & Assumptions

Given: The open set $U$ , the smooth function $F$ , and the derivative hypotheses in the statement.

[L1]

The Euclidean implicit function theorem solves one scalar equation for one variable as a smooth function of the remaining parameters when the relevant partial derivative is invertible (The Euclidean implicit function theorem with derivative formula).

[L2]

A Euclidean map with invertible derivative at a point is a local diffeomorphism there (The Euclidean inverse function theorem).

Proof

technique · local reduction

1.1

If $m = 0$ , define $A (u) : = \int_{0}^{1} (1 - t) F^{''} (t u) d t$ . Then $F (u) = u^{2} A (u)$ , $A (0) = \frac{1}{2} F^{''} (0) \neq 0$ , and after shrinking the domain one has $ε A (u) > 0$ for $ε = sgn (F^{''} (0))$ . Putting $u^{'} : = u \sqrt{ε A (u)}$ gives $F (u) = ε (u^{'})^{2}$ .

givenconstruct

1.2

Assume $m > 0$ . Put $G (u, y) : = \partial F / \partial u (u, y)$ . Since $\partial G / \partial u (0, 0) = \partial^{2} F / \partial u^{2} (0, 0) \neq 0$ , [L1] gives a smooth function $ϕ$ near $0 \in R^{m}$ with $ϕ (0) = 0$ and $G (ϕ (y), y) = 0$ . [L1, given, assume-case[ positive-parameter], construct]

2.1

Set $\tilde{F} (s, y) : = F (s + ϕ (y), y) - F (ϕ (y), y)$ . Then $\tilde{F} (0, y) = 0$ and $\partial \tilde{F} / \partial s (0, y) = 0$ for $y$ near $0$ .

step 1.2algebra

3.1

Define $A (s, y) : = \int_{0}^{1} (1 - t) \partial^{2} \tilde{F} / \partial s^{2} (t s, y) d t$ . The integral formula gives $\tilde{F} (s, y) = s^{2} A (s, y)$ , and $A (0, 0) = \frac{1}{2} \partial^{2} F / \partial u^{2} (0, 0) \neq 0$ . After shrinking, the sign $ε : = sgn (\partial^{2} F / \partial u^{2} (0, 0))$ satisfies $ε A (s, y) > 0$ everywhere.

step 2.1construct

4.1

Put $β (s, y) : = \sqrt{ε A (s, y)}$ , define $u^{'} : = β (s, y) s$ , and let $H (y) : = F (ϕ (y), y)$ . Then $F (s + ϕ (y), y) = H (y) + ε (u^{'})^{2}$ , and $\partial u^{'} / \partial s (0, 0) = β (0, 0) \neq 0$ , so [L2] makes $(s, y) \mapsto (u^{'}, y)$ a local diffeomorphism at $(0, 0)$ .

L2step 3.1construct

5.1

Composing the translation $(u, y) \mapsto (u - ϕ (y), y)$ from step 1.2 with the coordinate change from step 4.1 yields the required local coordinates $(u^{'}, y)$ , and the case $m = 0$ is already covered by step 1.1.

step 1.1step 1.2step 4.1∎

Depends on

Used by

Splitting one Morse coordinate preserves the residual Hessian Lemma
Morse lemma Theorem

Dependency tree · two levels

15 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)