Read-back: gray_stability_fixing_stationary_points

Statement. For all natural numbers $n, c$, every map $F : \mathbb{R}^n \to \mathbb{R}^c$, and every map $t \mapsto \alpha_t$ from $\mathbb{R}$ to one-forms on $\mathbb{R}^n$: if

• (hypothesis on $F$) $F$ is a compact regular level, i.e. $F \in C^\infty$, $M = F^{-1}(0)$ is compact, and $DF(y)$ is surjective for all $y \in M$; and
• (hypothesis on $\alpha$) $\alpha$ is a contact family on $M$, i.e. $(t,y) \mapsto \alpha_t(y)$ is $C^\infty$ on $\mathbb{R} \times \mathbb{R}^n$ and $\alpha_t$ is contact at $y$ for all $t \in [0,1]$ and all $y \in M$,

then there exists a map $\psi : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ which is an isotopy of $M$ (definition below) such that both of the following hold.

(A) Conformal relation. There exists $\lambda : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}$, $(t,y) \mapsto \lambda_t(y)$, which is $C^\infty$ on all of $\mathbb{R} \times \mathbb{R}^n$, with $\lambda_t(y) > 0$ (strictly) for all $t \in [0,1]$, $y \in M$, and

Positivity and the identity are required only on $[0,1] \times M$ (the identity only for $v \in T_y$); elsewhere $\lambda$ is only smooth. $\lambda_0 = 1$ is not required.

(B) Fixed points. For every point $p \in M$: if

then $\psi_t(p) = p$ for all $t \in [0,1]$.

Remarks on (B), as written:

• The same $\psi$ that witnesses (A) must satisfy (B), for all qualifying $p$ simultaneously ($\psi$ is chosen before $p$ is quantified; $\lambda$ plays no role in (B)).
• The hypothesis of (B) concerns only the single point $p$ (not a neighbourhood of $p$, and not $d\alpha$ or derivatives in $y$), only tangent vectors $v \in T_p$, and only times $t \in [0,1]$. The time derivative is the ordinary two-sided derivative of the real function $s \mapsto \alpha_s(p)(v)$ on $\mathbb{R}$ at $s = t$ (including at $t = 0$ and $t = 1$); since $\alpha$ is assumed smooth, the derivative exists and the "zero where not differentiable" convention does not come into play. Equivalently, the hypothesis says that $\alpha_t(p)$ restricted to $T_p$ does not depend on $t \in [0,1]$.
• The conclusion of (B) is only $\psi_t(p) = p$ for $t \in [0,1]$; nothing is asserted about $D\psi_t(p)$, about $\lambda_t(p)$, or about times outside $[0,1]$.
• If no point of $M$ satisfies the hypothesis of (B), then (B) is vacuous.

Setting and unfolded definitions. Throughout, $n, c \in \mathbb{N}$ are arbitrary natural numbers (both may be $0$), and $\mathbb{R}^n$ means the space of functions $\{0,\dots,n-1\} \to \mathbb{R}$ with the sup norm. A one-form $\alpha$ is any map $\mathbb{R}^n \to (\mathbb{R}^n)^*$, $y \mapsto \alpha_y$ (a continuous linear functional at each point; no regularity is built in). Derivatives $D$ are Fréchet derivatives of maps on the ambient space $\mathbb{R}^n$ (or $\mathbb{R} \times \mathbb{R}^n$), with the convention that a derivative is $0$ at any point where the map is not differentiable. For $F : \mathbb{R}^n \to \mathbb{R}^c$ and a one-form $\alpha$:

• Level set: $M := F^{-1}(0) = \{ y \in \mathbb{R}^n : F(y) = 0 \}$.

• Compact regular level (hypothesis on $F$): $F$ is $C^\infty$ on all of $\mathbb{R}^n$, $M$ is compact, and for every $y \in M$ the derivative $DF(y) : \mathbb{R}^n \to \mathbb{R}^c$ is surjective.

• Tangent space: $T_y := \ker DF(y) = \{ v \in \mathbb{R}^n : DF(y)\,v = 0 \}$ (defined for every $y$, used only for $y \in M$).

• Exterior derivative: $d\alpha_y(u,v) := \big(D\alpha(y)\,u\big)(v) - \big(D\alpha(y)\,v\big)(u)$, where $D\alpha(y) : \mathbb{R}^n \to (\mathbb{R}^n)^*$ is the ambient derivative of $y \mapsto \alpha_y$ (equal to $0$ if $\alpha$ is not differentiable at $y$).

• Contact plane: $\xi_y := \{ v \in T_y : \alpha_y(v) = 0 \}$.

• Contact at $y$: (i) there exists $v \in T_y$ with $\alpha_y(v) \neq 0$, and (ii) for every $u \in \xi_y$ with $u \neq 0$ there exists $v \in \xi_y$ with $d\alpha_y(u,v) \neq 0$ (i.e. $d\alpha_y$ restricted to $\xi_y \times \xi_y$ is nondegenerate; condition (ii) is vacuous when $\xi_y = \{0\}$). No orientation, sign, or dimension-parity condition is imposed explicitly.

• Smooth family: a map $t \mapsto \alpha_t$ from $\mathbb{R}$ to one-forms such that $(t,y) \mapsto \alpha_t(y) \in (\mathbb{R}^n)^*$ is $C^\infty$ on all of $\mathbb{R} \times \mathbb{R}^n$.

• Contact family on $M$ (hypothesis on $\alpha$): $\alpha$ is a smooth family, and for every $t \in [0,1]$ and every $y \in M$, $\alpha_t$ is contact at $y$ in the above sense. Nothing is required of $\alpha_t$ for $t \notin [0,1]$ beyond smoothness, nor off $M$ beyond smoothness.

• Isotopy of $M$: a map $\psi : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$, $(t,y) \mapsto \psi_t(y)$, such that

  1. $(t,y) \mapsto \psi_t(y)$ is $C^\infty$ on all of $\mathbb{R} \times \mathbb{R}^n$;
  2. $\psi_0(y) = y$ for every $y \in M$ (only on $M$, not on all of $\mathbb{R}^n$);
  3. for every $t \in [0,1]$: $\psi_t$ maps $M$ into $M$, is injective on $M$, and maps $M$ onto $M$; and for every $y \in M$, the linear map $D\psi_t(y)$ is injective on $T_y$.

  Nothing is required of $\psi_t$ for $t \notin [0,1]$ other than smoothness, and nothing off $M$ other than smoothness. The definition does not state separately that $D\psi_t(y)$ maps $T_y$ into $T_{\psi_t(y)}$.

• Pullback: $(\psi_t^*\alpha_t)_y(v) := \alpha_t\big(\psi_t(y)\big)\big(D\psi_t(y)\,v\big)$, where $D\psi_t(y)$ is the ambient derivative of $\psi_t : \mathbb{R}^n \to \mathbb{R}^n$ at $y$. Edge cases of the hypotheses.

• Surjectivity of $DF(y)$ at points of $M$ forces $\dim T_y = n - c$ there; if $c > n$ then $M$ must be empty.

• If $M = \emptyset$ (e.g. $c > n$; or $n = 0$ and $c \ge 1$; or $F$ a nonzero constant with $c \ge 1$), every condition quantified over $M$ is vacuous: both hypotheses reduce to smoothness of $F$ and of $\alpha$, and the conclusion only asks for smooth objects with no constraint coming from $M$.

• Contact condition (i) requires $T_y \neq \{0\}$ at each $y \in M$, so a nonempty $M$ needs $n - c \ge 1$. For $c = 0$ the target $\mathbb{R}^0$ is a point, so $F \equiv 0$ and $M = \mathbb{R}^n$; compactness forces $n = 0$, and then $T_y = \{0\}$ violates (i). Hence for $c = 0$ the two hypotheses cannot hold simultaneously.

• If $n - c = 1$, then $\xi_y = \{0\}$ and contact condition (ii) is vacuous; "contact" then only means that $\alpha_t(y)$ does not vanish identically on $T_y$.

• The contact condition and $d\alpha$ use the ambient derivative of $\alpha_t$ on $\mathbb{R}^n$, evaluated at points of $M$ on vectors of $\xi_y$.