Osaka Journal of Mathematics
- Osaka J. Math.
- Volume 44, Number 4 (2007), 765-793.
Hermitian structures on cotangent bundles of four dimensional solvable Lie groups
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  and , 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.
Osaka J. Math., Volume 44, Number 4 (2007), 765-793.
First available in Project Euclid: 7 January 2008
Permanent link to this document
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 17B30: Solvable, nilpotent (super)algebras 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 22E25: Nilpotent and solvable Lie groups
Secondary: 53C55: Hermitian and Kählerian manifolds [See also 32Cxx] 53D17: Poisson manifolds; Poisson groupoids and algebroids
de Andrés, Luis C.; Barberis, M. Laura; Dotti, Isabel; Fernández, Marisa. Hermitian structures on cotangent bundles of four dimensional solvable Lie groups. Osaka J. Math. 44 (2007), no. 4, 765--793. https://projecteuclid.org/euclid.ojm/1199719404