Associative submanifolds of the Berger space

Abstract

We study associative submanifolds of the Berger space $\mathrm{SO}(5)/\mathrm{SO}(3)$ endowed with its homogeneous nearly-parallel $\mathrm{G}_2$-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 $\mathrm{Gr}^+_2 \!\left( T S^4 \right)$. 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 $\mathrm{SO}(5)/\mathrm{SO}(3)$. An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial $\mathrm{SO}(3)$-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.