Connections on manifolds and new characteristic classes
type: Item
instance of: scholarly article
title: Connections on manifolds and new characteristic classes
zbMATH Open document ID: 0732.58018
DOI: 10.1007/BF00580851
author: Valerij V. Trofimov
published in: Acta Applicandae Mathematicae
publication date: 1991-01-01T00:00:00Z
review text: This paper contains some results announced in the author's note in Sov. Math., Dokl. 39, No.1, 214-217 (1989); translation from Dokl. Akad. Nauk SSSR 304, No.6, 1302-1305 (1989; Zbl 0687.58009)]. The author presents the construction and the properties of the characteristic classes of a submanifold \(N^ n\) in a manifold \(M^ m\) endowed with a linear connection. These are elements in \(H^*(N)\). For a characteristic class \(a\in H^ n(N^ n)\) it can be defined the index of \(N^ n\) corresponding to a. In the case where \((M^{2n},\omega)\) is a symplectic manifold endowed with a symplectic torsion free linear connection and \(N^ n\) is a Lagrangian submanifold it follows the definition of the Maslov classes and the generalized Maslov indices of \(N^ n\). In the case of the minimal Lagrangian submanifolds of a Kähler manifold the characteristic classes are zero. The author studies the cases of the spin manifolds, flat compact Riemannian manifolds, pseudo-Riemannian manifolds (mainly the Lorentz manifolds), and the de Rham decomposition theorem. The results are applied in studying the topological invariants of Hamiltonian systems, indices of Liouville tori, indices of the symplectic vector bundles, characteristic numbers of the isotropic submanifolds in a pseudo-Riemannian manifold.
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 4211109
zbMATH Keywords: Maslov class
zbMATH Keywords: Maslov index
zbMATH Keywords: affine connection
zbMATH Keywords: cobordism
zbMATH Keywords: holonomy group
zbMATH Keywords: minimal surface
zbMATH Keywords: characteristic classes
zbMATH Keywords: linear connection
zbMATH Keywords: symplectic manifold
zbMATH Keywords: Lagrangian submanifold
zbMATH Keywords: spin manifolds
zbMATH Keywords: flat compact Riemannian manifolds
zbMATH Keywords: pseudo-Riemannian manifolds
zbMATH Keywords: Lorentz manifolds
zbMATH Keywords: topological invariants of Hamiltonian systems
zbMATH Keywords: indices of Liouville tori
zbMATH Keywords: indices of the symplectic vector bundles
reviewed by: Vasile Oproiu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4211109