POSITIVELY CURVED MANIFOLDS WITH LARGE CONJUGATE RADIUS

2013 ◽  
Vol 05 (03) ◽  
pp. 333-344 ◽  
Author(s):  
BENJAMIN SCHMIDT
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Spaces ◽  
Injectivity Radius ◽  
Curved Manifolds ◽  
Rank One ◽  
Sectional Curvatures ◽  
Simply Connected ◽  
Positively Curved

Let M denote a complete simply connected Riemannian manifold with all sectional curvatures ≥1. The purpose of this paper is to prove that when M has conjugate radius at least π/2, its injectivity radius and conjugate radius coincide. Metric characterizations of compact rank one symmetric spaces are given as applications.

scholarly journals Positively Curved Riemannian Locally Symmetric Spaces are Positively Squared Distance Curved

2013 ◽  
Vol 65 (4) ◽  
pp. 757-767 ◽  
Author(s):  
Philippe Delanoë ◽  
François Rouvière
Keyword(s):  
Symmetric Space ◽  
Riemannian Manifolds ◽  
Symmetric Spaces ◽  
Direct Proof ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Hopf Fibrations ◽  
Rank One ◽  
Simply Connected ◽  
Positively Curved

AbstractThe squared distance curvature is a kind of two-point curvature the sign of which turned out to be crucial for the smoothness of optimal transportation maps on Riemannian manifolds. Positivity properties of that new curvature have been established recently for all the simply connected compact rank one symmetric spaces, except the Cayley plane. Direct proofs were given for the sphere, and an indirect one (via the Hopf fibrations) for the complex and quaternionic projective spaces. Here, we present a direct proof of a property implying all the preceding ones, valid on every positively curved Riemannian locally symmetric space.

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 Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

CLOSED MINIMAL HYPERSURFACES IN COMPACT SYMMETRIC SPACES

1992 ◽  
Vol 03 (05) ◽  
pp. 629-651 ◽  
Author(s):  
CLAUDIO GORODSKI
Keyword(s):  
Symmetric Spaces ◽  
Local Geometry ◽  
Minimal Hypersurfaces ◽  
Compact Symmetric Space ◽  
Orbital Geometry ◽  
Rank One ◽  
Minimal Cones ◽  
Area Minimizing ◽  
Simply Connected ◽  
Minimal Embedding

W.Y. Hsiang, W.T. Hsiang and P. Tomter conjectured that every simply-connected, compact symmetric space of dimension ≥4 must contain some minimal hypersurfaces of sphere type. With the aid of equivariant differential geometry, they showed that this is in fact the case for many symmetric spaces of rank one and two. Let M be one of the symmetric spaces: Sn(1)×Sn(1)(n≥4), SU(6)/Sp(3), E6/F4, ℍP2 (quaternionic proj. plane) or CaP2 (Cayley proj. plane). We prove the existence of infmitely many immersed, minimal hypersurfaces of sphere type in M which are invariant under a certain group G of isometries of M. Following Hsiang and the others, the equivariant method is also used here to reduce the problem to an investigation of geodesics in M/G equipped with a metric (with singularities) depending only on the orbital geometry of the transformation group (G, M). However, our constructions are based on area minimizing homogeneous cones, corresponding to a corner singularity of M/G with the local geometry of nodal type; this can be viewed as a variation of some of their constructions which depended on some unstable minimal cones of focal type. We further apply the equivariant method to construct a minimal embedding of S1×Sn−1×Sn−1 into Sn(1)×Sn(1)(n≥2) and a minimal, embedded hypersurface of sphere type in [Formula: see text], ℍPn×ℍPn (n≥2) and CaP2×CaP2.

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 Open manifolds with non-homeomorphic positively curved souls

2019 ◽  
Vol 169 (2) ◽  
pp. 357-376 ◽  
Author(s):  
DAVID GONZÁLEZ-ÁLVARO ◽  
MARCUS ZIBROWIUS
Keyword(s):  
Sectional Curvature ◽  
Existence Results ◽  
Positive Answer ◽  
Open Manifold ◽  
Positive Sectional Curvature ◽  
Open Manifolds ◽  
Curved Manifolds ◽  
Eschenburg Spaces ◽  
Simply Connected ◽  
Positively Curved

AbstractWe extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are given by explicit pairs of Eschenburg spaces. To deduce the second statement from the first, we extend our earlier work on the stable converse soul question and show that it has a positive answer for a class of spaces that includes all Eschenburg spaces.

scholarly journals A note on isometric immersions

1982 ◽  
Vol 33 (2) ◽  
pp. 162-166
Author(s):  
C. Baikoussis ◽  
F. Brickell
Keyword(s):  
Riemannian Manifold ◽  
Closed Ball ◽  
Isometric Immersions ◽  
Sectional Curvatures ◽  
Simply Connected

AbstractLet N be a complete connected Riemannian manifold with sectional curvatures bounded from below. Let M be a complete simply connected Riemannian manifold with sectional curvatures KM(σ)≤ −a2 (a ≥ 0) and with dimension < 2 dim N. Suppose that N is isometrically immersed in M and that its image lies in a closed ball of radius ρ. Then sup(KN(σ) − KM(σ)) ≥ μ2(aρ)/ρ2 where the function μ is defined by μ(x) = x coth x for x > 0, μ(0) = 1 and the supremum is taken over all sections tangent to N.

scholarly journals The Dirichlet problem at infinity on Hadamard manifolds

1995 ◽  
Vol 138 ◽  
pp. 1-18 ◽  
Author(s):  
Hironori Kumura
Keyword(s):  
Riemannian Manifold ◽  
Dirichlet Problem ◽  
Beltrami Operator ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Sectional Curvatures ◽  
Geodesic Rays ◽  
Simply Connected ◽  
Dirichlet Problem At Infinity

Let M be an n-dimensional Hadamard manifold, that is, a complete simply connected C∞ Riemannian manifold with nonpositive sectional curvatures. Making use of geodesic rays, Eberlein and O’Neill [11] constructed a compactification = MS(∞) of M which gives a homeomorphism of (M, S(∞)) with the Euclidean pair (Bn, Sn-1). In this paper we shall study the asymptotic Dirichlet problem for the Laplace-Beltrami operator, which is stated as follows:

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