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.

scholarly journals Non-Negative Curvature, Elliptic Genus and Unbounded Pontryagin Numbers

2018 ◽  
Vol 61 (2) ◽  
pp. 449-456
Author(s):  
Martin Herrmann ◽  
Nicolas Weisskopf
Keyword(s):  
Sectional Curvature ◽  
Negative Curvature ◽  
Elliptic Genus ◽  
Negatively Curved ◽  
Spin Manifolds ◽  
Simply Connected ◽  
Negative Sectional Curvature

AbstractWe discuss the cobordism type of spin manifolds with non-negative sectional curvature. We show that in each dimension 4k⩾ 12, there are infinitely many cobordism types of simply connected and non-negatively curved spin manifolds. Moreover, we raise and analyse a question about possible cobordism obstructions to non-negative curvature.

scholarly journals Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

2020 ◽  
Vol 63 (4) ◽  
pp. 901-908
Author(s):  
Philipp Reiser
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Positive Scalar ◽  
Space Of Riemannian Metrics ◽  
Simply Connected

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

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 AN ASSORTMENT OF NEGATIVELY CURVED ENDS

2013 ◽  
Vol 05 (04) ◽  
pp. 439-449 ◽  
Author(s):  
IGOR BELEGRADEK
Keyword(s):  
Sectional Curvature ◽  
Finite Volume ◽  
The Other ◽  
Negatively Curved ◽  
Negative Sectional Curvature

Motivated by recent groundbreaking work of Ontaneda, we describe a sizable class of closed manifolds such that the product of each manifold in the class with ℝ admits a complete metric of bounded negative sectional curvature which is an exponentially warped near one end and has finite volume near the other end.

scholarly journals Small covers, infra-solvmanifolds and curvature

2015 ◽  
Vol 27 (5) ◽  
Author(s):  
Shintarô Kuroki ◽  
Mikiya Masuda ◽  
Li Yu
Keyword(s):  
Sectional Curvature ◽  
Riemannian Metrics ◽  
Flat Torus ◽  
Combinatorial Structures ◽  
Simple Polytope ◽  
Simple Polytopes ◽  
Small Cover ◽  
Equivalent Conditions

AbstractIt is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover to be homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structures of the underlying simple polytopes.

scholarly journals Kuranishi-type moduli spaces for proper CR-submersions fibering over the circle

2019 ◽  
Vol 2019 (749) ◽  
pp. 87-132
Author(s):  
Laurent Meersseman
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Classical Case ◽  
Analytic Space ◽  
Fundamental Result ◽  
Compact Complex Manifold ◽  
Complex Structures ◽  
Analogous Statement ◽  
Finite Dimensional ◽  
Cr Structures

Abstract Kuranishi’s fundamental result (1962) associates to any compact complex manifold {X_{0}} a finite-dimensional analytic space which has to be thought of as a local moduli space of complex structures close to {X_{0}} . In this paper, we give an analogous statement for Levi-flat CR-manifolds fibering properly over the circle by associating to any such {\mathcal{X}_{0}} the loop space of a finite-dimensional analytic space which serves as a local moduli space of CR-structures close to {\mathcal{X}_{0}} . We then develop in this context a Kodaira–Spencer deformation theory making clear the likenesses as well as the differences with the classical case. The article ends with applications and examples.

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