Complexification and hypercomplexification of manifolds with a linear connection

Details Complexification and hypercomplexification of manifolds with a linear connection Complexification and hypercomplexification of manifolds with a linear connection

2002-12-12

arxiv.org/abs/math/0212175v2

Mathematics

Differential Geometry

some corrections, a reference added, to appear in International J. of Mathematics

Scientific paper

We give a simple interpretation of the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of $X$ in $TTX$, where $X$ is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperk\"ahler metric.

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