Connections on manifolds and new characteristic classes
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.
topological invariants of Hamiltonian systems
indices of the symplectic vector bundles
scientific article; zbMATH DE number 4211109
scientific article