Summary: Gwistor spaces
Departamento de Matemática da Universidade de Évora and Centro de Investigação em
Matemática e Aplicações (CIMA-UÉ), Rua Romão Ramalho, 59, 671-7000 Évora, Portugal
Abstract. We give a presentation of how one achieves the G2-twistor space of an oriented Rieman-
nian 4-manifold M. It consists of a natural SO(3) structure associated to the unit tangent sphere bun-
dle SM of the manifold. Many associated objects permit us to consider also a natural G2 structure.
We survey on the main properties of gwistor space and on recent results relating to characteristic
G2-connections with parallel torsion.
Keywords: metric connections, characteristic torsion, Einstein manifold, G2 structure
The construction of the G2-structure
Recall the exceptional Lie group G2 = AutO gives birth to a special geometry. A G2
structure on a Riemannian 7-manifold S is given by a stable 3-form , i.e.
· vector cross product · such that (X,Y,Z) = X ·Y,Z corresponds with the octo-
nionic product of pure imaginaries R7 O,
· = e456 + e014 + e025 + e036 - e126 - e234 - e315 on some oriented orthonormal co-
frame e0,...,e6, or
· lives in certain open GL(7)-orbit of 3T
u S .