Non-umbilical quaternionic contact hypersurfaces in hyper-K\"ahler manifolds

TL;DR

This paper investigates the geometric properties of quaternionic contact hypersurfaces within hyper-K"ahler manifolds, revealing conditions under which they are locally equivalent to the standard 3-Sasakian sphere.

Contribution

It establishes that non-umbilical compact qc hypersurfaces in hyper-K"ahler manifolds are locally qc homothetic to the 3-Sasakian sphere and describes the structure of nowhere umbilical qc hypersurfaces.

Findings

Non-umbilical compact qc hypersurfaces are locally qc homothetic to the 3-Sasakian sphere.

Nowhere umbilical qc hypersurfaces have an involutive 7-dimensional distribution.

Integral leaves of this distribution are locally qc-conformal to the 3-Sasakian sphere.

Abstract

We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-K\"ahler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-K\"ahler manifold is endowed with an involutive 7-dimensional distribution whose integral leafs are locally qc-conformal to the standard 3-Sasakian sphere.