Hermitian and quaternionic Hermitian structures on
March 15, 2007
We review the theory of quaternionic K¨ahler and hyperk¨ahler structures. Then
we consider the tangent bundle of a Riemannian manifold M with a metric con-
nection 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 K¨ahler 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.
Key Words: torsion, quaternionic, Hermitian, K¨ahler, symplectic.
MSC 2000: 53C15, 53C26, 53C55.
The author acknowledges the support of Funda¸c~ao Ci^encia e Tecnologia, through the
research center CIMA - Universidade de ´Evora and through the project POCI/MAT/-