 
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
