Parallelizable manifolds of type (1,1)

M.L. BARBERIS, I. DOTTI

Münster, Alemania

Congreso; International Conference on Global Differential Geometry; 2006

Universität Münster

Abelian complex structures on Lie groups have beenconsidered for the first time in the literature by Barberis-Dotti-Miatello. An abelian complex structureon a  Lie algebra $rak g$ is characterized by the followingcondition:egin{equation}label{1-1} ext{d} Lambda ^{1,0}rak g^* subset Lambda^{1,1}rak g^* . end{equation} Motivated by this property, wepropose a generalization of the concept of abelian complexstructures to parallelizable manifolds. We call such manifoldsemph{parallelizable manifolds of type $(1,1)$}. When $M$ is acomplex manifold of real dimension $2n$ and there exist $n$holomorphic vector fields $Z_1, dots , Z_n$, linearly independentat every point of $M$ then $M$ is called complex parallelizable. This notion, introduced by Wang, has been studied further by Nakamura. In this case, the holomorphic $1$-forms$alpha_1 , dots , alpha _n$ dual to the vector fields $Z_1, dots, Z_n$ satisfy the following condition, analogous to eqref{1-1}:$$ ext{d} alpha _i quad ext{is a section of }Lambda ^{2,0}M,quad i=1, dots , n .$$Therefore, according to our terminology, a complex parallelizablemanifold can be regarded as a parallelizable manifold of type$(2,0)$. We compare both notions and study conditions on aparallelizable manifold $M$ of type $(1,1)$  to be a quotient$M=Dackslash G$ of a Lie group $G$ by a discrete subgroup $D$ suchthat the complex structure on $M$ is induced from an abelian complexstructure on $G$.