scholarly journals Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank

2005 ◽  
Vol 332 (1) ◽  
pp. 81-101 ◽  
Author(s):  
Fuquan Fang ◽  
Xiaochun Rong
Keyword(s):  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Positively Curved ◽  
Symmetry Rank

scholarly journals Positively curved manifolds with maximal symmetry-rank

1994 ◽  
Vol 91 (1-3) ◽  
pp. 137-142 ◽  
Author(s):  
Karsten Grove ◽  
Catherine Searle
Keyword(s):  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Positively Curved ◽  
Symmetry Rank

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 Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

scholarly journals Positively curved manifolds with maximal discrete symmetry rank

2004 ◽  
Vol 126 (2) ◽  
pp. 227-245 ◽  
Author(s):  
Fuquan Fang ◽  
Xiaochun Rong
Keyword(s):  
Discrete Symmetry ◽  
Curved Manifolds ◽  
Positively Curved ◽  
Symmetry Rank

scholarly journals Flat strips, Bowen–Margulis measures, and mixing of the geodesic flow for rank one CAT(0) spaces

2015 ◽  
Vol 37 (3) ◽  
pp. 939-970 ◽  
Author(s):  
RUSSELL RICKS
Keyword(s):  
Geodesic Flow ◽  
Isometric Action ◽  
Structural Result ◽  
Curved Manifolds ◽  
Rank One ◽  
Unit Speed ◽  
Flat Strip ◽  
Geodesically Complete ◽  
Positively Curved ◽  
General Technical

Let$X$be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group$\unicode[STIX]{x1D6E4}$with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of$X$modulo the$\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in$\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when$X$has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when$X$is a tree with all edge lengths in$c\mathbb{Z}$for some$c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

scholarly journals Fibred coarse embedding into non-positively curved manifolds and higher index problem

2014 ◽  
Vol 267 (11) ◽  
pp. 4029-4065 ◽  
Author(s):  
Xiaoman Chen ◽  
Qin Wang ◽  
Zhijie Wang
Keyword(s):  
Curved Manifolds ◽  
Coarse Embedding ◽  
Index Problem ◽  
Positively Curved

scholarly journals Positively curved manifolds with symmetry

2006 ◽  
Vol 163 (2) ◽  
pp. 607-668 ◽  
Author(s):  
Burkhard Wilking
Keyword(s):  
Curved Manifolds ◽  
Positively Curved
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