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

TL;DR

This paper establishes necessary and sufficient conditions for the analytic regularizability of nondegenerate hypo-analytic structures of hypersurface type, linking it to associated CR submanifolds and regularizability of related structures.

Contribution

It introduces a new invariant CR submanifold called the central submanifold and connects the regularizability of the structure to that of this submanifold and related CR geometries.

Findings

Existence of a central CR submanifold determining regularizability.

Regularizability of the structure is equivalent to that of the central submanifold.

Regularizability of associated CR structures implies regularizability of the original structure.

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 $\Sigma$ 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 $\Sigma$. 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.