Being motivated by the study of semi-dispersing billiard systems [D. Burago, S. Ferleger, and A. Kononenko, Ergodic Theory Dyn. Syst. 18, No. 4, 791-805 (1998; Zbl 0922.58064)], the authors prove a theorem that can be formulated as follows.
Theorem. Let \(S\) be a 3-dimensional Euclidean simplex. Then it is possible to glue together finitely many copies \(S_j\), \(j=1,\dots, k\), of \(S\) so that (1) the \(S_j\)’s are glued together by identifying their faces isometrically in pairs and (2) the pseudomanifold without boundary \(\bigcup_{j=1,\dots ,k}S_j\) has nonpositive curvature in the sense of A. D. Alexandrov.
Recall that an \(n\)-dimensional pseudomanifold without boundary is a simplicial complex \(K\) such that (a) every simplex of \(K\) is a face of some \(n\)-simplex of \(K\); (b) every \((n-1)\)-dimensional simplex of \(K\) is the face of exactly two \(n\)-simplexes of \(K\); (c) if \(s\) and \(s'\) are \(n\)-simplexes of \(K\), there is a finite sequence \(s=s_1, s_2, \dots , s_m=s'\) of \(n\)-simplexes of \(K\) such that \(s_j\) and \(s_{j+1}\) have an \((n-1)\)-face in common for \(1<j<m\), see, for example, [E. H. Spanier, Algebraic topology. New York etc.: McGraw-Hill Book Company (1966; Zbl 0145.43303)].
The authors claim that, in the above theorem, the simplex \(S\) may be replaced by any 3-dimensional smooth compact nonpositively curved Riemannian manifold with corners, such that its boundary consists of a finite number of geodesically convex nonpositively curved faces and all the corners are nondegenerate (i.e., all the one and two-dimensional angles an the corners are nonzero).
The proof uses Thurston’s Geometrization Theorem for Haken manifolds that may be found in the recent book by M. Kapovich [Hyperbolic manifolds and discrete groups. Boston: Birkhäuser (2001; Zbl 0958.57001)]. No version of the main theorem is given in dimension \(\geq 4\).