The G_2 sphere over a 4-manifold

Details The G_2 sphere over a 4-manifold The G_2 sphere over a 4-manifold

2006-08-11

arxiv.org/abs/math/0608282v1

Monatshefte f\"ur Mathematik, Vol 158, Number 4 December 2009, pp. 335-348

Mathematics

Differential Geometry

13 pages

Scientific paper

10.1007/s00605-008-0053-3

We present a construction of a canonical G_2 structure on the unit sphere tangent bundle S_M 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.

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