From almost (para)-complex structures to affine structures on Lie groups

Abstract

Let G= H⋉ K denote a semidirect product Lie group with Lie algebra g= h⊕ k, where k is an ideal and h is a subalgebra of the same dimension as k. There exist some natural split isomorphisms S with S2= ± Id on g: given any linear isomorphism j: h→ k, we get the almost complex structure J(x, v) = (- j- 1v, jx) and the almost paracomplex structure E(x, v) = (j- 1v, jx). In this work we show that the integrability of the structures J and E above is equivalent to the existence of a left-invariant torsion-free connection ∇ on G such that ∇ J= 0 = ∇ E and also to the existence of an affine structure on H. Applications include complex, paracomplex and symplectic geometries.