Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

Details Hermitian structures on cotangent bundles of four dimensional solvable Lie groups Hermitian structures on cotangent bundles of four dimensional solvable Lie groups

2006-04-27

arxiv.org/abs/math/0604608v2

Osaka J. Math. 44, 765--793, 2007

Mathematics

Differential Geometry

26 pages. Typos corrected

Scientific paper

We study hermitian structures, with respect to the standard neutral metric on the cotangent bundle $T^*G$ of a 2n-dimensional Lie group $G$, which are left invariant with respect to the Lie group structure on $T^*G$ induced by the coadjoint action. These are in one-to-one correspondence with left invariant generalized complex structures on $G$. Using this correspondence and results of Cavalcanti-Gualtieri and Fern\'andez-Gotay-Gray, it turns out that when $G$ is nilpotent and four or six dimensional, the cotangent bundle $T^*G$ always has a hermitian structure. However, we prove that if $G$ is a four dimensional solvable Lie group admitting neither complex nor symplectic structures, then $T^*G$ has no hermitian structure or, equivalently, $G$ has no left invariant generalized complex structure.

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