Summary: SOLVMANIFOLDS WITH INTEGRABLE AND NON­INTEGRABLE
G 2 STRUCTURES
ILKA AGRICOLA, SIMON G. CHIOSSI, AND ANNA FINO
Abstract. We show that a 7­dimensional non­compact Ricci­flat Riemannian man­
ifold with Riemannian holonomy G2 can admit non­integrable G2 structures of type
R#S 2
0 (R 7 ) #R 7 in the sense of Fern’andez and Gray. This relies on the construction of
some G2 solvmanifolds, whose Levi­Civita connection is known to give a parallel spinor,
admitting a 2­parameter family of metric connections with non­zero skew­symmetric
torsion that has parallel spinors as well. The family turns out to be a deformation
of the Levi­Civita connection. This is in contrast with the case of compact scalar­flat
Riemannian spin manifolds, where any metric connection with closed torsion admitting
parallel spinors has to be torsion­free.
1. Introduction
The study and explicit construction of Riemannian metrics with holonomy G 2 on non­
compact manifolds of dimension seven (called metrics with parallel or integrable G 2
structure) has been an exciting area of di#erential geometry since the pioneering work
of Bryant and Salamon in the second half of the eighties (cf. [Br87], [BrS89] and [Sa89]).
Mathematical elegance aside, these metrics have turned out to be an important tool in
superstring theory, since they are exact solutions of the common sector of type II string

Source: Agricola, Ilka - Institut für Mathematik, Humboldt-Universität zu Berlin

Collections: Mathematics