A remark on the intersections of subanalytic leaves (Q2809651)

Let \(M\) and \(N\) denote two \(C^1\)-smooth subanalytic leaves in \({\mathbb R}^n\). Given a point \(a\in M\cap N\), the condition \(C_a(M\cap N)=T_aM\cap T_aN\), where \(C_a\) denotes the tangent cone at \(a\), is generally not sufficient for \((M\cap N)_a\) to be a \(C^1\)-smooth germ. The author investigates additional conditions ensuring smoothness. For instance, if \(\max\{\dim M,\dim N\}\leq 2\), then the aforementioned condition is sufficient for \((M\cap N)_a\) to be smooth. An identity principle for real-analytic submanifolds is also established, as well as for subanalytic functions: if \(f:({\mathbb R}^n,0)\to ({\mathbb R},0)\) is a \(C^1\) subanalytic germ with analytic graph such that \(C_0(f^{-1}(\{0\}))={\mathbb R}^n\), then \(f=0\).