On the Gauss-Bonnet formula for locally symmetric spaces of noncompact type (Q1355387)

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.