On the Whitney extension property for continuously differentiable horizontal curves in sub-Riemannian manifolds

Abstract

In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $C^1$ extension property. We conclude by showing that the $C^1$ extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.

Appendix

Let us present here a useful technical result based on standard topological degree considerations.

Lemma 5.5

Let V be a normed space and W an affine and dense subspace of V. Fix $v\in V$ and let ${\mathcal {F}}\in C^1(V,\mathbb R^d)$ be a submersion at v. Consider a sequence of functions ${\mathcal {F}}_n:W\rightarrow \mathbb R^d$ with the property that ${\mathcal {F}}_n$ locally uniformly converges to ${\mathcal {F}}|_{W}$. Let, moreover, $z_n$ be a sequence in $\mathbb R^d$ converging to ${\mathcal {F}}(v)$. Then there exists a sequence $w_n$ in W converging to v in V and such that, for n large enough, ${\mathcal {F}}_n(w_n)=z_n$.

Proof

By assumption there exist $\phi _1,\dots ,\phi _d\in V$ such that the map

$$\begin{aligned} \begin{array}{rccl} {\mathfrak {F}}: &{} \mathbb R^d &{} \longrightarrow &{} \mathbb R^d \\ &{} (x_1,\dots ,x_d) &{} \longmapsto &{} {\mathcal {F}}(v+x_1 \phi _1+\dots +x_d\phi _d) \end{array} \end{aligned}$$

is a local diffeomorphism at 0.

Let $v_n$ be a sequence in W converging to v in V. Denote by $W_L$ the linear space $\{w-w'\mid w,w'\in W\}$ and consider, for each $i=1,\dots ,d$, a sequence $\varphi _i^n$ in $W_L$ converging to $\phi _i$ in V. Then the sequence of maps

$$\begin{aligned} \begin{array}{rccl} {\mathfrak {G}}_n: &{} \mathbb R^d &{} \longrightarrow &{} \mathbb R^d \\ &{} (x_1,\dots ,x_d) &{} \longmapsto &{} {\mathcal {F}}_n(v_n+x_1 \phi _1^n+\dots +x_d\phi _d^n) \end{array} \end{aligned}$$

locally uniformly converges to ${\mathfrak {F}}$.

Let $r>0$ be small enough so that the restriction of ${\mathfrak {F}}$ to the ball $B_r(0)$ of center the origin and radius r is a diffeomorphism between $B_r(0)$ and ${\mathfrak {F}}(B_r(0))$. Then ${\mathfrak {G}}_n|_{\overline{B_r(0)}}$ uniformly converges to ${\mathfrak {F}}|_{\overline{B_r(0)}}$. Hence, for any K compactly contained in ${\mathfrak {F}}(B_r(0))$ and for n large enough, the topological degree $d({\mathfrak {G}}_n,B_r(0),z)$ is equal to 1 or $-1$ for every $z\in K$. In particular, choosing $K={\mathfrak {F}}(\overline{B_{\rho r}(0)})$ with $\rho \in (0,1/2)$ and replacing r by $2\rho r$ in the above argument, we have that for n large enough, there exists $x^n\in B_{2\rho r}(0)$ such that

$$\begin{aligned} {\mathcal {F}}_n(v_n+x_1^n \phi _1^n+\dots +x_d^n\phi _d^n)={\mathfrak {G}}_n(x^n)=z_n. \end{aligned}$$

In order to recover the convergence to v of the sequence $v_n+x_1^n \phi _1^n+\dots +x_d^n\phi _d^n$ it suffices to notice that if $z_n$ is in ${\mathfrak {F}}(\overline{B_{\rho r}(0)})$, then $x^n$ can be chosen of norm smaller than $2\rho r$. The conclusion then follows from the convergence of $v_n$ to v and the uniform boundedness of $\{\phi _j^n\mid j=1,\dots ,d,\,n\in \mathbb N\}$. $\square$