Hermitian and quaternionic Hermitian structures on tangent bundles

Details Hermitian and quaternionic Hermitian structures on tangent bundles Hermitian and quaternionic Hermitian structures on tangent bundles

2007-03-15

arxiv.org/abs/math/0703450v1

Geometriae Dedicata, Vol. 133, 1, April 2008, pp. 95-110

Mathematics

Differential Geometry

18 pages

Scientific paper

10.1007/s10711-008-9237-1

We review the theory of quaternionic Kahler and hyperkahler structures. Then we consider the tangent bundle of a Riemannian manifold M with a metric connection D (with torsion) and with its well estabilished canonical complex structure. With an extra almost Hermitian structure on M it is possible to find a quaternionic Hermitian structure on TM, which is quaternionic Kahler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM. Finally a proper S^3-bundle of complex structures is introduced, expanding to TM the well known twistor bundle of M.

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