The G2 sphere over a 4-manifold
I. M. C. Salavessa2
We present a construction of a canonical G2 structure on the unit sphere tangent bundle
SM of any given orientable Riemannian 4-manifold M. Such structure is never geometric or
1-flat, but seems full of other possibilities. We start by the study of the most basic properties of
our construction. The structure is co-calibrated if, and only if, M is an Einstein manifold. The
fibres are always associative. In fact, the associated 3-form results from a linear combination
of three other volume 3-forms, one of which is the volume of the fibres. We also give new
examples of co-calibrated structures on well known spaces. We hope this contributes both to
the knowledge of special geometries and to the study of 4-manifolds.
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.