Conformally parallel G2 structures on a class of solvmanifolds

Abstract

Abstract. Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G 2 -metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G 2 structure. By suitably deforming the SU(3) structures obtained, we are able to describe the corresponding nonhomogeneous Ricci-flat metrics with holonomy contained in G 2 . In the process we also find a new metric with exceptional holonomy.

Special Symplectic Six-Manifolds

“…One of these is satisfied automatically: indeed, taking d of dω 3 It is now clear that σ − 3 can be expressed in terms of t, giving (24). Using the general formula…”

Locally conformal parallel $G_2$ and Spin(7) manifolds

“…More recently, conformal parallel G 2 structures have been studied in details on solvmanifolds [CF04]. A locally conformal parallel structure is encoded in the Lee form θ, the 1-form representing the irreducible component of the covariant derivative of the fundamental form in the standard representation.…”

Conformally parallel Spin(7) structures on solvmanifolds

“…The classification of these types of manifolds is an ongoing problem, also being treated by the authors of [16]. Finally, we mentioned different directions on G 2 and Spin(7) manifolds related to the geometric structures on these spaces.…”

“…Inspired by this work [16], moving up one dimension, we study Spin(7) structures on a rank-one solvable extension of a metric 7 -dimensional nilpotent Lie algebra n endowed with an G 2 structure φ and a nonsingular self-adjoint derivation D , which is diagonalizable by a unitary basis in order to obtain the noncompact examples found in [25].…”