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Differential Geometry and its Applications
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Differential Geometry and its Applications
Article . 2014 . Peer-reviewed
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Invariant pseudo-Kähler metrics on generalized flag manifolds

Authors: Yamada, Takumi;

Invariant pseudo-Kähler metrics on generalized flag manifolds

Abstract

A pseudo-Riemannian manifold \((M,g)\) with a complex structure \(J\) is called a \textit{pseudo-Kähler manifold} if \(g\) is a pseudo-Hermitian metric and the fundamental \(2\)-form \(\omega _{g}\) is closed. Hence, a pseudo-Kähler manifold is one of the natural generalizations of a Kähler manifold. Such a pseudo-Riemannian metric \(g\) is called \textit{pseudo-Kähler metric} and the fundamental \(2\)-form \(\omega _{g}\) \textit{pseudo-Kähler structure}. In this interesting paper, the author considers signatures of invariant pseudo-Kähler metrics on generalized flag manifolds from the viewpoint of \(T\)-root systems. By a result of Maschler, to investigate the signatures of the integral pseudo-Kähler metrics of a generalized flag manifold, it is sufficient to investigate the signatures of the invariant pseudo-Kähler metrics. By investigating patterns of signatures of invariant pseudo-Kähler metrics on a generalized flag manifold, the author gets that two invariant complex structures \(J\) and \(J^{\prime }\) on a generalized flag manifold are either diffeomorphic or non-diffeomorphic.

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Keywords

pseudo-Kähler metric, Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics, Differential geometry of homogeneous manifolds, generalized flag manifolds, Global differential geometry of Hermitian and Kählerian manifolds, \(T\)-root system

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
4
Average
Average
Average
hybrid
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