A characteristic class for totally real surfaces in the Grassmannian of two-planes in four-space
type: Item
instance of: scholarly article
title: A characteristic class for totally real surfaces in the Grassmannian of two-planes in four-space
zbMATH Open document ID: 0599.32025
DOI: 10.1007/BF00132255
author: Thomas Fiedler
published in: Annals of Global Analysis and Geometry
publication date: 1986-01-01T00:00:00Z
review text: Let \(Q_ 2=G_{4,2}({\mathbb{R}})\) be the non-degenerate quadric in \({\mathbb{P}}^ 3({\mathbb{C}})\) identified with the real Grassmann manifold of planes in \({\mathbb{R}}^ 4\). Let G denote the Grassmann bundle associated to the real tangent space \(TQ_ 2\) and let G' denote the sub-bundle of G corresponding fibre-wisely to the totally real planes in \(TQ_ 2\). If M is a totally real surface embedded into \(Q_ 2\), then its Gauss map \(t: M\to G\) factorizes through G'. Due to the topology of G', the manifold M carries by pull-back a form \(\alpha_ M\in H^ 1_{comp}(M,{\mathbb{Z}}_ 2)\) which reflects its position in G'. This cohomology class is analogous to the Maslov-Arnold index attached to Lagrangian submanifolds of \({\mathbb{R}}^{2n}.\) The author presents an invariant construction of the class \(\alpha_ M\) in terms of the twistor space. As an application of these tools, the author studies the geometry of algebraic curves in \({\mathbb{P}}^ 3({\mathbb{C}})\), via the twistor map \({\mathbb{P}}^ 3({\mathbb{C}})\to S^ 4\).
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 51M35
zbMATH DE Number: 3966685
zbMATH Keywords: twistor transformation
zbMATH Keywords: Grassmann manifold
zbMATH Keywords: totally real surface
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00132255
OpenAlex ID: W1968504277
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description: scientific article; zbMATH DE number 3966685