Sur la géométrie systolique des variétés de Bieberbach |
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Authors: | Chady El Mir Jacques Lafontaine |
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Institution: | (1) Institut de Mathématiques et Modélisation de Montpellier CNRS, UMR 5149, Université Montpellier 2, Case 51, 34095 Montpellier Cedex 5, France |
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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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Keywords: | Torsion Quaternionic Hermitian K?hler Symplectic |
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