Let $G = H\ltimes K$ denote a semidirect product Lie group with Lie algebra $g = h \oplus k$, where $k$ is an ideal and $h$ is a subalgebra of the same dimension as $k$. There exist some natural split isomorphisms $S$ with $S^2 = \pm Id$ on $g$: given any linear isomorphism $j : h \to k$, we get the almost complex structure $J (x, v) = (−j^{−1}v, j x)$ and the almost paracomplex structure $E(x, v) = ( j^{−1}v, j x)$. 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 $\nabla$ on $G$ such that$ \nabla J = 0 = \nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.
From almost (para)-complex structures to affine structures on Lie groups
Calvaruso, Giovanni;
2018-01-01
Abstract
Let $G = H\ltimes K$ denote a semidirect product Lie group with Lie algebra $g = h \oplus k$, where $k$ is an ideal and $h$ is a subalgebra of the same dimension as $k$. There exist some natural split isomorphisms $S$ with $S^2 = \pm Id$ on $g$: given any linear isomorphism $j : h \to k$, we get the almost complex structure $J (x, v) = (−j^{−1}v, j x)$ and the almost paracomplex structure $E(x, v) = ( j^{−1}v, j x)$. 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 $\nabla$ on $G$ such that$ \nabla J = 0 = \nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
