Hermitian and quaternionic Hermitian structures on tangent bundles
نویسنده
چکیده
We review the theory of quaternionic Kähler and hyperkähler 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 Kähler if, and only if, D is flat and torsion free. We also review the symplectic nature of TM . Finally a proper S3-bundle of complex structures is introduced, expanding to TM the well known twistor bundle of M .
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