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Summary: arXiv:0808.1714v2[math.DG]24Nov2009
On the G2 bundle of a Riemannian 4-manifold
R. Albuquerque
rpa@dmat.uevora.pt
November 24, 2009
Abstract
We study the natural G2 structure on the unit tangent sphere bundle SM of any
given orientable Riemannian 4-manifold M, as it was discovered in [3]. A name is
proposed for the space. We work in the context of metric connections, or so called
geometry with torsion, and describe the components of the torsion of the connection
which imply certain equations of the G2 structure. This article is devoted to finding
the G2-torsion tensors which classify our structure according to the theory in [10].
Contents
1 Recalling the theory 2
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 The unit tangent sphere bundle, its metric and Levi-Civita connection . . . 3
2 The natural G2 structure on SM 6
2.1 Gwistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The structure equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 The trivial gwistor space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
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