A version of implicit function theorem when sections are not everywhere smooth?

Question

Let $V_1, V_2 \rightarrow M $ be smooth vector bundles over a manifold $M$ and $s_1: M \rightarrow V_1$ a smooth section transverse to the zero set and $s_2: M \rightarrow V_2$ a continuous section such that $ s_2 : s_1^{-1}(0) \rightarrow V_2$ is a smooth section, transverse to the zero set. Let $p\in M$ be a point such that $$ s_1(p)=0, \qquad s_2(p) =0. $$ Does there always exist a solution $p(t_1, t_2)$ near $p$ such that $$ s_1(p(t_1,t_2)) =t_1 \qquad s_2(p(t_1,t_2)) = t_2 $$ if $t_1$ and $t_2$ are sufficiently small? To make sense of sufficiently small we can just chose some metric on $V_1$ and $V_2$.

Note that if I said that the section $p \rightarrow s_1(p) \oplus s_2(p)$ is smooth and transverse to the zero set, then the answer is obviously yes (by implicit function theorem). It seems that one should be able to imitate the proof of implicit function theorem (ie contraction mapping principal) to justify what I am asking. But I am not sure of this.

Answer 1

Note that by your assumption $s_2$ is only smooth along the submanifold $s^{-1}(0)$ of $M$ and might even jump in directions transverse to it. So it is easy to come up with a counterexample: $M=\mathbb R^2$, $s_1(x,y)=x$, $s_2(x,y) = y$ for $x\le 0$ and $=y+1$ for $x>0$.

But if you assume that $s_2$ is strictly differentiable on $M$ along $s_1^{-1}(0)$, then is true in a somewhat weaker sense. See

• MR0817719 Reviewed Ver Eecke, Paul Applications du calcul différentiel. (French) [Applications of differential calculus] Mathématiques. [Mathematics] Presses Universitaires de France, Paris, 1985. 397 pp.