scholarly journals Smooth and PL-rigidity problems on locally symmetric spaces

2016 ◽  
Vol 7 (4) ◽  
pp. 205
Author(s):  
Ramesh Kasilingam
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Metrics ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Topological Properties ◽  
Open Problems ◽  
Locally Symmetric Spaces ◽  
Negatively Curved ◽  
Rigidity Problems

This is a survey on known results and open problems about smooth and PL-rigidity problem for negatively curved locally symmetric spaces. We also review some developments about studying the basic topological properties of the space of negatively curved Riemannian metrics and the Teichmuller space of negatively curved metrics on a manifold.

scholarly journals Teichmüller space of negatively curved metrics on Gromov–Thurston manifolds is not contractible

2014 ◽  
Vol 06 (04) ◽  
pp. 541-555 ◽  
Author(s):  
Gangotryi Sorcar
Keyword(s):  
Symmetric Space ◽  
Homotopy Type ◽  
Riemannian Metrics ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Locally Symmetric Space ◽  
Branched Cover ◽  
Negative Sectional Curvature

In this paper we prove that for all n = 4k - 2, k ≥ 2 there exists closed n-dimensional Riemannian manifolds M with negative sectional curvature that do not have the homotopy type of a locally symmetric space, such that [Formula: see text] is nontrivial. [Formula: see text] denotes the Teichmüller space of all negatively curved Riemannian metrics on M, which is the topological quotient of the space of all negatively curved metrics modulo the space of self-diffeomorphisms of M that are homotopic to the identity. Gromov–Thurston branched cover manifolds provide examples of negatively curved manifolds that do not have the homotopy type of a locally symmetric space. These manifolds will be used in this paper to prove the above stated result.

Some Borsuk-Ulam-type theorems for maps from Riemannian manifolds into manifolds

1989 ◽  
Vol 111 (1-2) ◽  
pp. 61-67
Author(s):  
Duan Hai-bao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Topological Properties ◽  
Locally Symmetric Spaces ◽  
Cut Points ◽  
Continuous Map ◽  
Ulam Theorem

SynopsisSuppose f: M →N is a continuous map from a Riemannian manifold (M, d) into a manifold N. The main result of this paper is to give some conditions under which f identifies a pair of cut points. This result leads to generalisations of the classical Borsuk-Ulam theorem. As a consequence some topological properties of locally symmetric spaces are discovered.

Riemannian metrics on Teichmüller space

1996 ◽  
Vol 89 (1) ◽  
pp. 281-306 ◽  
Author(s):  
Lutz Habermann ◽  
Jürgen Jost
Keyword(s):  
Riemannian Metrics ◽  
Teichmüller Space ◽  
Teichmuller Space

A dynamical-geometric characterization of the geodesic flows of negatively curved locally symmetric spaces

2014 ◽  
Vol 35 (7) ◽  
pp. 2094-2113
Author(s):  
YONG FANG
Keyword(s):  
Symmetric Spaces ◽  
Geodesic Flows ◽  
Anosov Flow ◽  
Rigidity Result ◽  
Locally Symmetric Spaces ◽  
Negatively Curved ◽  
Finite Covers ◽  
Constant Change ◽  
Rank One

In this paper we prove the following rigidity result: let ${\it\varphi}$ be a $C^{\infty }$ topologically mixing transversely symplectic Anosov flow. If (i) its weak stable and weak unstable distributions are $C^{\infty }$ and (ii) its Hamenstädt metrics are sub-Riemannian, then up to finite covers and a constant change of time scale, ${\it\varphi}$ is $C^{\infty }$ flow conjugate to the geodesic flow of a closed locally symmetric Riemannian space of rank one.

scholarly journals THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES

2004 ◽  
Vol 06 (03) ◽  
pp. 419-430 ◽  
Author(s):  
M. T. MUSTAFA
Keyword(s):  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Positive Curvature ◽  
Compact Manifolds ◽  
Harmonic Morphisms ◽  
Compact Type ◽  
Riemannian Submersions ◽  
Locally Symmetric Spaces ◽  
Totally Geodesic ◽  
Negatively Curved

The structure of local and global harmonic morphisms between Riemannian manifolds, with totally fibres, is investigated. It is shown that non-positive curvature of the domain obstructs the existence of global harmonic morphisms with totally geodesic fibres and the only such maps from compact Riemannian manifolds of non-positive curvature are, up to a homothety, totally geodesic Riemannian submersions. Similar results are obtained for local harmonic morphisms with totally geodesic fibres from open subsets of non-negatively curved compact and non-compact manifolds. During the course, we prove non-existence of submersive harmonic morphisms with totally geodesic fibres from some important domains, for instance from compact locally symmetric spaces of non-compact type and open subsets of symmetric spaces of compact type.

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