Associative Submanifolds of the Berger Space

TL;DR

This paper investigates associative submanifolds within the Berger space with a nearly-parallel G2-structure, establishing correspondences with pseudo-holomorphic curves and classifying special Gauss map cases, revealing new homogeneous examples.

Contribution

It introduces a correspondence between ruled associatives and pseudo-holomorphic curves, proves the existence of infinitely many topological types of compact associatives, and classifies those with special Gauss maps with non-trivial stabilizers.

Findings

Correspondence with pseudo-holomorphic curves in $Gr_2^+(TS^4)$

Existence of infinitely many topological types of compact associatives

Classification of associatives with special Gauss maps with non-trivial stabilizers

Abstract

We study associative submanifolds of the Berger space SO(5)/SO(3) endowed with its homogeneous nearly-parallel G2-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map. We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $Gr_2^+(TS^4).$ Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds in SO(5)/SO(3). An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial SO(3)-stabiliser. We classify the associative submanifolds with special Gauss map in the cases where the stabiliser contains an element of order greater than 2. In particular, we find several homogeneous examples of this type.