Calibrated Submanifolds of R^7 and R^8 with Symmetries

Abstract

The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in R^7 and R^8 are given.

21 pages, minor corrections, section added containing further systems of differential equations defining calibrated submanifolds

Cited by in corpus (8)

• Ruled Lagrangian Submanifolds of the 6-Sphere
• Associative Submanifolds of the 7-Sphere
• Bryant-Salamon ${\mathrm{G}}_2$ manifolds and coassociative fibrations
• Conjectures on counting associative 3-folds in $G_2$-manifolds
• Asymptotically Conical Associative 3-folds
• Cayley fibrations in the Bryant-Salamon $Spin(7)$ manifold
• Deformed $G_2$-instantons on ${\mathbb{R}}^4\times S^3$
• Deformations of calibrated subbundles in noncompact manifolds of special holonomy via twisting by special sections