On the Gauss-Bonnet formula for locally symmetric spaces of noncompact type
type: Item
instance of: scholarly article
title: On the Gauss-Bonnet formula for locally symmetric spaces of noncompact type
zbMATH Open document ID: 0871.22006
author: Enrico Leuzinger
published in: L'Enseignement Mathématique. 2e Série
publication date: 1997-09-24T00:00:00Z
review text: Let \(X\) be a noncompact Riemannian symmetric space of \(\text{rank} \geq 2\), and \(\Gamma\) an irreducible nonuniform, torsion free lattice in the group of isometries of \(X\). \textit{G. Harder} [Ann. Sci. Éc. Norm. Sup., IV. Sér. 4, 409-455 (1971; Zbl 0232.20088)] first proved the Gauss-Bonnet formula for a locally symmetric space \(V= \Gamma \backslash X\). He constructed a smooth exhaustion function \(h\) on \(V\), whose sublevel sets have complicated geometric properties; hence Harder's proof involves rather long and technical estimates. The approach of this paper is based on an exhaustion \(V= \cup_{s\geq 0} V(s)\) of locally symmetric space by polyhedra, i.e., compact Riemannian manifolds with corners. The essential feature of this exhaustion is that it is naturally related to the geometry of \(V\), e.g., the second fundamental forms of the boundaries of \(\partial V(s)\) are uniformly bounded. This together with the generalized Gauss-Bonnet formula for Riemannian polyhedra give a much simpler proof of the Gauss-Bonnet theorem for locally symmetric spaces.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 1013723
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: lattice
zbMATH Keywords: Gauss-Bonnet formula
zbMATH Keywords: locally symmetric space
zbMATH Keywords: Riemannian polyhedra
reviewed by: Fuliu Zhu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1013723