Showing papers on "Hyperbolic manifold published in 1986"
TL;DR: In this article, for a variete hyperbolique orientee fermee X and ρ a representation orthogonale de Π 1 X, a fonction zeta de type Ruelle Rρ(s) is defined for Re(s), i.e., le produit sur les geodesiques closes premieres γ des facteurs det [I−ρ(γ) exp(−sl(γ))], ou l(γ est la longueur de
Abstract: Pour une variete hyperbolique orientee fermee X et ρ une representation orthogonale de Π 1 X, on definit une fonction zeta de type Ruelle Rρ(s) pour Re(s) grand comme le produit sur les geodesiques closes premieres γ des facteurs det [I−ρ(γ) exp(−sl(γ))], ou l(γ) est la longueur de γ. On continue analytiquement Rρ(s) et on calcule son terme principal en s=0
238 citations
TL;DR: In this article, the authors take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations.
Abstract: In this paper we take a first step toward understanding representations of cocompact lattices in SO(n,1) into arbitrary Lie groups by studying the deformations of rational representations — see Proposition 5.1 for a rather general existence result. This proposition has a number of algebraic applications. For example, we remark that such deformations show that the Margulis Super-Rigidity Theorem, see [30], cannot be extended to the rank 1 case. We remark also that if Γ ⊂ SO(n,1) is one of the standard arithmetic examples described in Section 7 then Γ has a faithful representation ρ′ in SO(n+1), the Galois conjugate of the uniformization representation, and Proposition 5.1 may be used to deform the direct sum of ρ′ and the trivial representation in SO(n+2) thereby constructing non-trivial families of irreducible orthogonal representations of Γ. However, most of this paper is devoted to studying certain spaces of representations which are of interest in differential geometry in a sense which we now explain.
217 citations
TL;DR: In this paper, a real hypersurface of a complex hyperbolic space #x2102 was constructed and a principal circle bundle over it was constructed, which is a Lorentzian hypergraph of the anti-De Sitter space H12n+1.
Abstract: Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.
184 citations
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42 citations
TL;DR: In this article, the authors present a Symmetric hyperbolic system with a large parameter and discuss the communication in Partial Differential Equations (PDE) for the system.
Abstract: (1986). Symmetric hyperbolic systems with a large parameter. Communications in Partial Differential Equations: Vol. 11, No. 15, pp. 1627-1651.
42 citations
TL;DR: In this article, the authors exploit the close interplay between hyperbolic 3-manifolds and pseudoAnosov maps which arises via the mapping torus construction to prove results in the theory of surface automorphisms which appear to be difficult to obtain directly.
29 citations
TL;DR: Soit M une variete hyperbolique a n dimensions de volume fini V, and soit λ 1 le premier element positif dans le spectre discret pour le probleme Δf+λf=0 sur M as discussed by the authors.
Abstract: Soit M une variete hyperbolique a n dimensions de volume fini V, et soit λ 1 le premier element positif dans le spectre discret pour le probleme Δf+λf=0 sur M. On considere des bornes inferieures pour λ 1 dans le cas de volume fini
25 citations
TL;DR: In this article, the relevance of compact hyperbolic manifolds in the context of Kaluza-Klein theories is discussed, and examples of spontaneous compactification on hyper-bolic manifold, including d -dimensional (d ⩾8) Einstein-Yang-Mills gravity and 11-dimensional supergravity are considered.
01 Jan 1986
TL;DR: In this paper, the authors discuss the Cauchy problem for uniformly diagonalizable hyperbolic systems of linear partial differential equations in Gevrey classes and show that the uniform diagonalizability of these systems is a sufficient condition for strong hyper-bolicity.
Abstract: Publisher Summary This chapter discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems of linear partial differential equations in Gevrey classes. It also discusses the Cauchy problem for uniformly diagonalizable hyperbolic systems whose coefficients are in Gevrey class. It has been proven that if the coefficients of the system are constant, the uniformly diagonalizable hyperbolic system is equivalent to strongly hyperbolic one, that is, this system is stable under the perturbation of lower order term of the system. In general, the characterization of the strong hyperbolicity in the C ∞ -sense for the systems of variable coefficients is an open problem. The uniform diagonalizability for the hyperbolic systems is a sufficient condition for the Cauchy problem to be well-posed.
01 Mar 1986
TL;DR: A manifold M is said to be aspherical if its universal covering space is contractible as discussed by the authors, i.e., it is a manifold that is topologically rigid.
Abstract: A manifold M is said to be aspherical if its universal covering space is contractible. Farrell and Hsiang have conjectured [3]: Conjecture A. (Topological rigidity of aspherical manifolds.) Any homotopy equivalence f: N → M between closed aspherical manifolds is homotopic to a homeomorphism, and its analogue in algebraic K -theory: Conjecture B . The Whitehead groups Wh j (π 1 M)(j ≥ 0) of the fundamental group of a closed aspherical manifold M vanish.
TL;DR: In this article, a non-compact polyhedra for complete and incomplete hyperbolic orbifolds is studied and a strong rigidity result is proved for the complete case in dimension exceeding two.
Abstract: This paper deals with filling the hyperbolic space Hn by non-compact polyhedra. In dimensions n <4 the non-compact case is very different from the compact one, which was investigated by A.D. Aleksandrov. For n ≥ 4 the compact and non-compact cases are “almost” similar. This investigation is closely related to deformations of complete and incomplete hyperbolic orbifolds (in the sense of W. Thurston) for which a strong rigidity result is proved-similar to the one for complete hyperbolic manifolds in dimension exceeding two.
TL;DR: In this paper, it was shown that all complete Riemannian manifolds with constant negative sectional curvature and finite volume are admissible if they can be computed in terms of stable concordance spaces.
Abstract: Let r = TT\\M where M is a complete hyperbolic manifold with finite volume. We announce (among other results) that Wh T = 0 where Wh T is the Whitehead group of T. We also announce WI12 r = 0, k0(ZT) = 0, K-n(ZT) = 0 (for n > 0), and Whn T 0 Q = 0 (for all n). We calculate the weak homotopy type of the stable topological concordance space C(M), and hence Waldhausen's Wh-theory (cf. [22]) of M, in terms of simpler stable concordance spaces. When M is compact, the calculation is in terms of ^(S) where S is the circle. A connected complete Riemannian manifold M is called weakly admissible if there exist positive real numbers a < b such that all the sectional curvatures of M are less than —a and bigger than —6. A weakly admissible manifold is admissible if it has finite volume. In particular, all complete locally symmetric spaces having finite volume and strictly negative sectional curvatures are admissible Riemannian manifolds. These are precisely the real, complex, quaternionic and Cayley complete hyperbolic manifolds of finite volume. All complete manifolds of constant negative sectional curvature and finite volume occur among these; in fact, they are the complete real hyperbolic manifolds of finite volume. The purpose of this paper is to announce the calculation of the algebraic K-theory of admissible manifolds. We start by stating that the Whitehead group Wh TT\\M of the fundamental group of an admissible manifold M vanishes. Actually, we proceed to formulate and state a bit more general result. A group T is K-flat if Wh(r©C) = 0 for all nonnegative integers n where C denotes the free abelian group of rank n. The Bass-Heller-Swan formula [3] implies WhT = 0, K0{ZT) = 0 and i t_ n (Zr ) = 0 provided T is If-flat and n > 0. A smooth fiber bundle F —• E ~ M is admissible if