Complexification and Hypercomplexification of Manifolds with a Linear Connection

We give a simple interpretation of the adapted complex structure of Lempert-Szőke 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 hyperkähler metric. Let X be a manifold equipped with a linear connection ∇. Then the tangent bundle TX of X has a canonical foliation (nonsingular on TX\X) by tangent bundles to geodesics, i.e. by surfaces Tγγ, where γ is a geodesic. A complex structure on TX (or some neighbourhood of X) is called adapted (to ∇) if the leaves of the canonical foliation are complex (immersed) submanifolds of TX . For Riemannian connections the adapted complex structures were constructed and studied by R. Szőke and L. Lempert [10, 7]. An equivalent definition was given by V. Guillemin and M. Stenzel [3]. The results of Lempert and Szőke can be formulated as follows: Theorem 0.1. [7, 10, 3] Let (X, g) be a Riemannian manifold. There exists a (unique) adapted complex structure for the Levi-Civita connection on some neighbourhood of X in TX if and only if (X, g) is a real-analytic Riemannian manifold (i.e. both X and g are real-analytic). In this paper we shall give a simple construction of an adapted complex structure which works for any real-analytic manifold X with real-analytic linear connection ∇. It is sort of a polar decomposition of the complexified manifold and generalises the basic example of the adapted complex structure on TG, where G is a compact Lie group equipped with the bi-invariant metric. Actually, this construction is implicitly used by R. Szőke [10] in the proof that his (different) construction yields the adapted complex structure. In the Riemannian case, (X, g), once we have the adapted complex structure J on TX , we obtain a Kähler metric on TX whose Kähler form is the canonical 2-form on the tangent bundle of a Riemannian manifold. This metric has in particular the property of being flat on leaves of the canonical foliation. 2000 Mathematics Subject Classification. 53C26. Research supported by an EPSRC Advanced Research Fellowship.