Abelian Hermitian geometry

2011/06/30 by Andrada, Adrian, Barberis, Maria Laura, Dotti, Isabel · 1 citation
#53B35 #53C15 #53C30 #Differential Geometry (math.DG) #FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1106.6268

Abstract

We study the structure of Lie groups admitting left invariant abelian complex structures in terms of commutative associative algebras. If, in addition, the Lie group is equipped with a left invariant Hermitian structure, it turns out that such a Hermitian structure is Kähler if and only if the Lie group is the direct product of several copies of the real hyperbolic plane by a euclidean factor. Moreover, we show that if a left invariant Hermitian metric on a Lie group with an abelian complex structure has flat first canonical connection, then the Lie group is abelian.

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• Positive Hermitian curvature flow on 2-step nilpotent Lie groups

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• Complex structures on two-step nilpotent Lie groups
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