Hermitian and quaternionic Hermitian structures on tangent bundles

Abstract

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.

18 pages

References in corpus (1)

• Twistor theory of symplectic manifolds

Cited by in corpus (2)

• On vector bundle manifolds with spherically symmetric metrics
• A fundamental differential system of 3-dimensional Riemannian geometry