Associative submanifolds of the Berger Space

$\def\SO{\mathrm{SO}}$We study associative submanifolds of the Berger space $\SO(5) / \SO(3)$ 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 pseudoholomorphic curves in $\mathrm{Gr}^+_2 (TS)^4$. 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.

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Received 9 April 2020

Accepted 15 April 2021

Published 16 July 2024