Coclosed G2-structures on SU(2)2-invariant cohomogeneity one manifolds

Abstract

We consider two different SU(2)2-invariant cohomogeneity one manifolds, one non-compact M=R4×S3 and one compact M=S4×S3, and study the existence of coclosed SU(2)2-invariant G2-structures constructed from half-flat SU(3)-structures. For R4×S3, we prove the existence of a family of coclosed (but not necessarily torsion-free) G2-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed G2-structure constructed from a half-flat SU(3)-structure is in this family. For S4×S3, we prove that there are no SU(2)2-invariant coclosed G2-structures constructed from half-flat SU(3)-structures.