 
Summary: arXiv:math.DG/0608282v111Aug2006
The G2 sphere over a 4manifold
R. Albuquerque1
I. M. C. Salavessa2
rpa@uevora.pt isabel.salavessa@ist.utl.pt
Abstract
We present a construction of a canonical G2 structure on the unit sphere tangent bundle
SM of any given orientable Riemannian 4manifold M. Such structure is never geometric or
1flat, but seems full of other possibilities. We start by the study of the most basic properties of
our construction. The structure is cocalibrated if, and only if, M is an Einstein manifold. The
fibres are always associative. In fact, the associated 3form results from a linear combination
of three other volume 3forms, one of which is the volume of the fibres. We also give new
examples of cocalibrated structures on well known spaces. We hope this contributes both to
the knowledge of special geometries and to the study of 4manifolds.
Key Words: connections on principal bundles, sphere bundle, G2 structure, Einstein
manifold, spin bundle, holonomy group.
MSC 2000: Primary: 53C10, 53C20, 53C25; Secondary: 53C05, 53C28
The authors acknowledge the support of Funda¸c~ao Ci^encia e Tecnologia, either
through the project POCI/MAT/60671/2004 and through their research centers, re
spectively, CIMA and CFIF.
