Scalar differential invariants and characteristic classes of homogeneous geometrical structures
type: Item
instance of: scholarly article
title: Scalar differential invariants and characteristic classes of homogeneous geometrical structures
zbMATH Open document ID: 0814.57019
author: D. M. Gessler
author: Alexander Verbovetsky
author: A. M. Vinogradov
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: It was pointed out in [\textit{A. M. Vinogradov}, Scalar differential invariants, diffieties and characteristic classes, in: `Mechanics, analysis and geometry: 200 years after Lagrange, 379-414 (1991; Zbl 0735.57012)], the relationship between the algebra of scalar differential invariants of homogeneous geometrical structures and their characteristic classes, namely, that the characteristic classes are the cohomology classes of the regular \(\mathbb{R}\)-spectrum of the corresponding algebra of differential invariants. In the present paper the next step is taken up and the cohomology of the above-mentioned regular \(\mathbb{R}\)-spectrum is computed. Namely, it is shown that this cohomology coincides with the cohomology of the classifying space BG of the subgroup \(G\) of the general differential group \(\mathbb{G}^ p(n)\) that defines the relevant geometrical structure of order \(p\). For example, the characteristic classes of pseudo-Riemannian metrics of type \((\ell,m)\) are exhausted by the cohomology classes \(H^ i(\text{BSO}(\ell,m))\), \(0\leq i\leq \ell + m\), and the characteristic classes of their \(s\)-dimensional bordisms by the cohomology classes \(H^{\ell + m + s} (\text{BSO}(\ell,m))\), \(s > 0\).
Mathematics Subject Classification ID: 57R15
Mathematics Subject Classification ID: 57R20
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 525688
zbMATH Keywords: cohomology of regular \(\mathbb{R}\)-spectrum
zbMATH Keywords: algebra of scalar differential invariants
zbMATH Keywords: homogeneous geometrical structures
zbMATH Keywords: characteristic classes
zbMATH Keywords: classifying space
reviewed by: Vladimir L. Popov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 525688