Examples of hyper-Kähler connections with torsion. (English) Zbl 0989.53028

Poon, Yat Sun

Marchiafava, S. (ed.) et al., Proceedings of the 2nd meeting on quaternionic structures in mathematics and physics, Roma, Italy, September 6-10, 1999. Dedicated to the memory of André Lichnerowicz and Enzo Martinelli. Rome: Dipartimento di Matematica “Guido Castelnuovo”, Università di Roma “La Sapienza”, 321-327, electronic only (2001).

On a hyper-Hermitian manifold there are two natural torsion-free connections, namely the Levi-Civita connection and the Obata connection. However, in general the Levi-Civita connection does not preserve the hypercomplex structure and the Obata connection does not preserve the metric. A linear connection \(\nabla\) on a hyper-Hermitian manifold is hyper-Hermitian if \(\nabla g=0\) and \(\nabla I_1=\nabla I_2=\nabla I_3\), where \(I_1, I_2,I_3\) form the hypercomplex structure. Note that a general hyper-Hermitian connection has torsion. In this paper the author presents some examples of HKT-connections (hyper-Kähler with torsion). A holomorphic characterization of HKT-structure is also given.
For the entire collection see [Zbl 0958.00032].

Reviewer: Marian Munteanu (Iaşi)

MSC:

53C26

Hyper-Kähler and quaternionic Kähler geometry, “special” geometry

Keywords:

hypercomplex geometry; hyper-Kähler with torsion; hyper-Hermitian manifold; Levi-Civita connection; Obata connection; HKT-connections