Cohomogeneity One Coassociative Submanifolds in the Bundle of Anti-self-dual 2-forms over the 4-sphere

Abstract

Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_2$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in ${\Lambda }_{-}^2{S}^4$, which is the complete $G_2$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of ${\Lambda }_{-}^2{S}^4$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical singularities and their desingularizations.

arXiv comment36 pages, v3: minor corrections, published version

Cited by in corpus (2)

• Bryant-Salamon ${\mathrm{G}}_2$ manifolds and coassociative fibrations
• Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold