Exercise 30, § 5

Let $v= (v_1, . . . , v_n):\mathbf{R}^n\rightarrow \mathbf{R}^n$ be a mapping of class $C^1$ and let $\delta$ be a real number $>0$; we make the following assumptions:

– The support of $v$ is contained in $[-\delta ,1 +\delta ]\times \mathring{\mathbf{B}}_{n-1}$;

– For every $x\in [0,1-\delta [$, one has $v_1(x,0, . . . ,0)>0$, and $v_1(1,0, . . . ,0) = 0$;

– For every $x\in [0,1]$, one has $v_2(x,0, . . . ,0) =\cdots =v_n(x,0, . . . ,0) = 0$. Let $\Phi :\mathbf{R}^n\times \mathbf{R}\rightarrow \mathbf{R}^n$ be the integral flow of the mapping $(x, t)\mapsto v(x)$ (VAR, 9.1.3); for $t\in \mathbf{R}$, we denote by $\Phi_t$ the mapping $x\mapsto \Phi (x, t)$.

a) Prove that there exists a real number $\tau$ such that $\Phi_{\tau}([0,1]\times \{0\})\subset [1-$ $\delta ,1]\times \{0\}$.

b) Let $j: [-\delta ,1 +\delta ]\times \mathbf{B}_{n-1}\rightarrow \mathbf{R}^n$ be a continuous injective mapping. For every $t\in [0,1]$, put $A_t=j([t,1]\times \{0\})$. Prove that, for every $t\in ]0,1[$, the subspaces $A_0$ and $A_t$ are isotopic.

c) Let P be a subset of $\mathbf{R}^n$ homeomorphic to $[0,1]$ which is the union of a finite family of segments. Prove that P is isotopic to a segment.