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Title: Special Kaehler manifolds: A survey (English)
Author: Cortés, Vincente
Language: English
Journal: Proceedings of the 21st Winter School "Geometry and Physics"
Issue: 2001
Pages: [11]-18
Category: math
Summary: This is a survey of recent contributions to the area of special K\"ahler geometry. A (pseudo-)K\"ahler manifold $(M,J,g)$ is a differentiable manifold endowed with a complex structure $J$ and a (pseudo-)Riemannian metric $g$ such that i) $J$ is orthogonal with respect to the metric $g,$ ii) $J$ is parallel with respect to the Levi Civita connection $D.$ A special K\"ahler manifold $(M,J,g,\nabla)$ is a K\"ahler manifold $(M,J,g)$ together with a flat torsionfree connection $\nabla$ such that i) $\nabla \omega = 0,$ where $\omega = g(.,J.)$ is the K\"ahler form and ii) $\nabla$ is symmetric. A holomorphic immersion $\phi : M \rightarrow V$ is called K\"ahlerian if $\phi^{\star} \gamma$ is nondegenerate and it is called Lagrangian if $\phi^{\star}\Omega= 0.$\par Theorem 1. Let $\phi:M \rightarrow V$ be a K\"ahlerian Lagrangian immersion with induced geometric data $(g,\nabla).$ Then $(M,J,g,\nabla)$ is a special K\"ahler manifold. Conversely, any simply connected sp! (English)
MSC: 32Q15
MSC: 32Q20
MSC: 53C26
MSC: 53C55
idZBL: Zbl 1039.53079
idMR: MR1972422
Date available: 2009-07-13T21:46:31Z
Last updated: 2012-09-18
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