Closed geodesics in simply connected Riemannian spaces of negative curvature

2000 ◽  
Vol 41 (5) ◽  
pp. 880-883
Author(s):  
V. K. Ionin
Keyword(s):  
Negative Curvature ◽  
Closed Geodesics ◽  
Simply Connected ◽  
Riemannian Spaces

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

scholarly journals AN INFINITE NUMBER OF CLOSED FLRW UNIVERSES FOR ANY VALUE OF THE SPATIAL CURVATURE

2012 ◽  
Vol 18 ◽  
pp. 77-80
Author(s):  
HELIO V. FAGUNDES
Keyword(s):  
Finite Volume ◽  
Infinite Number ◽  
Negative Curvature ◽  
Large Scale ◽  
Positive Curvature ◽  
Cosmological Models ◽  
Infinite Volume ◽  
Spatial Curvature ◽  
Multiply Connected ◽  
Simply Connected

The Friedman-Lemaître-Robertson-Walker cosmological models are based on the assumptions of large-scale homogeneity and isotropy of the distribution of matter and energy. They are usually taken to have spatial sections that are simply connected; they have finite volume in the positive curvature case, and infinite volume in the null and negative curvature ones. I want to call the attention to the existence of an infinite number of models, which are based on these same metrics, but have compact, finite volume, multiply connected spatial sections. Some observational implications are briefly mentioned.

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.

Lower bounds for the first Dirichlet eigenvalue of the Laplacian for domains in hyperbolic space

2015 ◽  
Vol 160 (2) ◽  
pp. 191-208 ◽  
Author(s):  
SERGEI ARTAMOSHIN
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Negative Curvature ◽  
Connected Space ◽  
Constant Negative Curvature ◽  
Dirichlet Eigenvalue ◽  
Dirichlet Eigenvalues ◽  
First Dirichlet Eigenvalue ◽  
A New Technique ◽  
Simply Connected

AbstractWe consider domains in a simply connected space of constant negative curvature and develop a new technique that improves existing classical lower bound for Dirichlet eigenvalues obtained by H. P. McKean as well as the lower bounds recently obtained by A. Savo.

EXPLICIT CONSTRUCTIONS OF UNIVERSAL ℝ-TREES AND ASYMPTOTIC GEOMETRY OF HYPERBOLIC SPACES

2001 ◽  
Vol 33 (6) ◽  
pp. 727-734 ◽  
Author(s):  
ANNA DYUBINA ◽  
IOSIF POLTEROVICH
Keyword(s):  
Free Group ◽  
Negative Curvature ◽  
Axiom Of Choice ◽  
Hyperbolic Spaces ◽  
Asymptotic Cone ◽  
Connected Manifold ◽  
Explicit Constructions ◽  
The Axiom Of Choice ◽  
Asymptotic Geometry ◽  
Simply Connected

This paper presents explicit constructions of universal ℝ-trees as certain spaces of functions, and also proves that a 2ℵ0-universal ℝ-tree can be isometrically embedded at infinity into a complete simply connected manifold of negative curvature, or into a non-abelian free group. In contrast to asymptotic cone constructions, asymptotic spaces are built without using the axiom of choice.

scholarly journals On a construction of complete simply-connected riemannian manifolds with negative curvature

1989 ◽  
Vol 113 ◽  
pp. 7-13
Author(s):  
Haruo Kitahara ◽  
Hajime Kawakami ◽  
Jin Suk Pak
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Negative Curvature ◽  
Harmonic Forms ◽  
Simply Connected

Let M be a complete simply-connected riemannian manifold of even dimension m. J. Dodziuk and I.M. Singer ([D1]) have conjectured that H2p(M) = 0 if p ≠ m/2 and dim H2m/2(M) = ∞, where H2*(M) is the space of L2-harmonic forms on M.

scholarly journals Graphs and metric 2-step nilpotent Lie algebras

2018 ◽  
Vol 18 (3) ◽  
pp. 265-284 ◽  
Author(s):  
Rachelle C. DeCoste ◽  
Lisa DeMeyer ◽  
Meera G. Mainkar
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Inner Product ◽  
Nilpotent Lie Group ◽  
Closed Geodesics ◽  
Nilpotent Lie Algebra ◽  
Comprehensive Description ◽  
Nilpotent Lie Algebras ◽  
Simply Connected ◽  
Dense Set

AbstractDani and Mainkar introduced a method for constructing a 2-step nilpotent Lie algebra 𝔫G from a simple directed graph G in 2005. There is a natural inner product on 𝔫G arising from the construction. We study geometric properties of the associated simply connected 2-step nilpotent Lie group N with Lie algebra 𝔫g. We classify singularity properties of the Lie algebra 𝔫g in terms of the graph G. A comprehensive description is given of graphs G which give rise to Heisenberg-like Lie algebras. Conditions are given on the graph G and on a lattice Γ ⊆ N for which the quotient Γ \ N, a compact nilmanifold, has a dense set of smoothly closed geodesics. This paper provides the first investigation connecting graph theory, 2-step nilpotent Lie algebras, and the density of closed geodesics property.

Sign in / Sign up

Export Citation Format

Share Document