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Hermitian and quaternionic Hermitian structures on tangent bundles

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Abstract

We review the theory of quaternionic Kähler and hyperkähler structures. Then we consider the tangent bundle of a Riemannian manifold M endowed with a metric connection D, with torsion, and with its well estabilished canonical complex structure. With an 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, in the wider context of geometry with torsion. Finally we discover an S 3-bundle of complex structures, which expands to TM the well known S 2-twistor bundle of a quaternionic Hermitian manifold M.

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  1. Departamento de Matemática, Centro de Investigação em Matemática e Aplicações (CIMA), Universidade de Évora, Rua Romão Ramalho, 59, 7000, Évora, Portugal

    Rui Albuquerque

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  1. Rui Albuquerque
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Correspondence to Rui Albuquerque.

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Albuquerque, R. Hermitian and quaternionic Hermitian structures on tangent bundles. Geom Dedicata 133, 95–110 (2008). https://doi.org/10.1007/s10711-008-9237-1

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