Gromov's K-area and symplectic rigidity
type: Item
instance of: scholarly article
title: Gromov's K-area and symplectic rigidity
zbMATH Open document ID: 0873.58037
DOI: 10.1007/BF02247119
published in: Geometric and Functional Analysis. GAFA
publication date: 1997-11-02T00:00:00Z
full work available at URL: https://eudml.org/doc/58246
review text: The author presents several results about a new symplectic invariant introduced by M. Gromov. The ingredients are: a closed symplectic manifold \((M^{2n},\Omega)\), a complex vector bundle \(E\to M\) with Hermitian metric \(h\), an \(h\)-preserving connection \(\nabla\) and a \(\Omega\)-compatible almost complex structure \(J\) on \(M\). The latter means that \(g(\xi,\eta)= \Omega(\xi,J\eta)\) is a Riemannian metric. Gromov defined the quantity \[ K\text{-area} (M,\Omega)= \sup_E k^E(M,\Omega) \] where \(E\) ranges over the (homologically significant) vector bundles. Here \[ k^E(M,\Omega)= \sup_{J,h,\nabla}|R^\nabla |^{-1}_{J,H} \quad\text{and}quad |R^\nabla|_{J,h}= \sup_{(\xi,\eta),v}|h(R^\nabla (\xi,\eta)v,v)| \] with \(v\in E_{(\cdot)}\) \(h\)-unit vector and \((\xi,\eta)\) in \(T_{(\cdot)}M\) \(g\)-orthonormal. The author discusses symplectic rigidity type results for the \(K\)-area with respect to individual vector bundles.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 934962
zbMATH Keywords: Gromov area
zbMATH Keywords: symplectic topology
zbMATH Keywords: symplectic rigidity
author: Leonid Polterovich
reviewed by: Jair Koiller
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 934962