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.

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.

Harmonic functions on ℝ-covered foliations

2009 ◽  
Vol 29 (4) ◽  
pp. 1141-1161
Author(s):  
S. FENLEY ◽  
R. FERES ◽  
K. PARWANI
Keyword(s):  
Harmonic Functions ◽  
Riemannian Metrics ◽  
Related Result ◽  
Continuous Functions ◽  
Property A ◽  
Liouville Property ◽  
Foliated Manifold ◽  
Curved Manifolds ◽  
Codimension One ◽  
Negatively Curved

AbstractLet (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

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

On a conjecture of Green

1997 ◽  
Vol 17 (1) ◽  
pp. 247-252
Author(s):  
CHENGBO YUE
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Geodesic Flow ◽  
Rigidity Theorem ◽  
Locally Symmetric Space ◽  
Closed Riemannian Manifold ◽  
Rank One ◽  
The Mean ◽  
Negative Sectional Curvature

Green [5] conjectured that if $M$ is a closed Riemannian manifold of negative sectional curvature such that the mean curvatures of the horospheres through each point depend only on the point, then $V$ is a locally symmetric space of rank one. He proved this in dimension two. In this paper we prove that under Green's assumption, $M$ must be asymptotically harmonic and that the geodesic flow on $M$ is $C^{\infty}$ conjugate to that of a locally symmetric space of rank one. Combining this with the recent rigidity theorem of Besson–Courtois–Gallot [1], it follows that Green's conjecture is true for all dimensions.

scholarly journals On the topology of moduli spaces of non-negatively curved Riemannian metrics

2021 ◽  
Author(s):  
Wilderich Tuschmann ◽  
Michael Wiemeler
Keyword(s):  
Sectional Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Rational Homotopy ◽  
Analogous Statement ◽  
Open Manifolds ◽  
Negatively Curved ◽  
Space Of Riemannian Metrics ◽  
Negative Sectional Curvature ◽  
Infinite Sequences

AbstractWe study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist infinite sequences of closed (respectively, open) manifolds of pairwise distinct homotopy type for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.

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