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arXiv:math.DG/0703450v115Mar2007 Hermitian and quaternionic Hermitian structures on

Summary: arXiv:math.DG/0703450v115Mar2007
Hermitian and quaternionic Hermitian structures on
tangent bundles
Rui Albuquerque
March 15, 2007
We review the theory of quaternionic Kšahler and hyperkšahler 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 Kšahler 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, Kšahler, symplectic.
MSC 2000: 53C15, 53C26, 53C55.
The author acknowledges the support of Fundažc~ao Ci^encia e Tecnologia, through the
research center CIMA - Universidade de ŽEvora and through the project POCI/MAT/-


Source: Albuquerque, Rui - Departamento de Matemática, Universidade de Évora


Collections: Mathematics