On the regularity of nondegenerate hypo-analytic structures of hypersurface type

Abstract

For a smooth, non-degenerate locally integrable structure of hypersurface type on a manifold $M$, we provide necessary and sufficient conditions for it to be equivalent, near a point, to a real-analytic locally integrable structure (the analytic regularizability), generalizing a recent result of Zaitsev and the first author. First, we discover, in our setting, a (previously unknown) invariant CR submanifold $Σ$ in $M$ of hypersurface type, which we call the central submanifold. We prove that the analytic regularizability of $M$ is equivalent to that of the associated CR manifold $Σ$. Furthermore, as a byproduct of our construction, we show that the central manifold construction reduces the whole (smooth or analytic) equivalence problem for nondegenerate structures with the Levi positivity condition to that of the associated central manifolds, i.e. to CR geometry. Second, we make use of a classical construction due to Marson and show that sufficient for the analytic regularizability of $M$ is the analytic regularizability of the CR manifold $\tilde M$ associated with $M$ in the sense of Marson. We show applications of both regularizability conditions to classes of locally integrable structures.