Regular interval diffeomorphism

Normalized gradient crosses a compact regular band in controlled time: Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $f:M\to \mathbb{R}$ be smooth on a boundaryless manifold, $a<b$, and let $K=f^{-1}([a,b])$ be compact with $df\mathrm{\ne }0$ on $K$. For any Riemannian metric there is a compactly supported smooth field $Y$ agreeing with $\mathrm{grad}f\mathrm{/}\mathrm{\parallel }\mathrm{grad}f{\mathrm{\parallel }}^2$ near $K$. Its complete flow $\mathrm{\Phi }$ satisfies $f({\mathrm{\Phi }}_t(x))=f(x)+t$ for $x\in K$ and $a-f(x)\le t\le b-f(x)$. Thus every intervening level is reached in exactly its value difference.

The fundamental theorem on flows: Let $X$ be a smooth vector field on $M$. For each $p\in M$, let ${\gamma }_p:I_p\to M$ be the maximal integral curve through $p$, and set $\mathcal{D}:=\{(t,p)\in \mathbb{R}\times M:t\in I_p\},\mathrm{\Phi }(t,p):={\gamma }_p(t)\mathrm{.}$ Then $\mathcal{D}$ is open in $\mathbb{R}\times M$, each fibre ${\mathcal{D}}_p$ is an interval containing $0$, the map $\mathrm{\Phi }:\mathcal{D}\to M$ is smooth, and $\mathrm{\Phi }$ is the unique maximal local flow generated by $X$.