scholarly journals On the growth of fundamental groups of nonpositive curvature manifolds

1996 ◽  
Vol 54 (3) ◽  
pp. 483-487 ◽  
Author(s):  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Group ◽  
Sectional Curvature ◽  
Exponential Growth ◽  
Compact Riemannian Manifold ◽  
Nonpositive Curvature ◽  
Fundamental Groups ◽  
Positive Sectional Curvature ◽  
Classic Result ◽  
Negative Sectional Curvature

Milnor's classic result that the fundamental group of a compact Riemannian manifold of negative sectional curvature has exponential growth is generalised to the case of negative Ricci curvature and non-positive sectional curvature.

scholarly journals Almost Positive Curvature on an Irreducible Compact Rank 2 Symmetric Space

2018 ◽  
Vol 2020 (5) ◽  
pp. 1346-1365 ◽  
Author(s):  
Jason DeVito ◽  
Ezra Nance
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Lie Group ◽  
Positive Curvature ◽  
Positive Sectional Curvature ◽  
Compact Symmetric Space ◽  
Rank 2 ◽  
Set Of Points ◽  
Positively Curved

Abstract A Riemannian manifold is said to be almost positively curved if the set of points for which all two-planes have positive sectional curvature is open and dense. We show that the Grassmannian of oriented two-planes in $\mathbb{R}^{7}$ admits a metric of almost positive curvature, giving the first example of an almost positively curved metric on an irreducible compact symmetric space of rank greater than 1. The construction and verification rely on the Lie group $\mathbf{G}_{2}$ and the octonions, so do not obviously generalize to any other Grassmannians.

Rigidities of asymptotically Euclidean manifolds

1999 ◽  
Vol 60 (3) ◽  
pp. 521-528 ◽  
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Group ◽  
Compact Riemannian Manifold ◽  
Universal Covering ◽  
Covering Space ◽  
Universal Covering Space

Let M be an n-dimensional compact Riemannian manifold. We study the fundamental group of M when the universal covering space of M, M is close to some Euclidean space ℝs asymptotically.

scholarly journals On the p-pseudoharmonic map heat flow

2020 ◽  
Vol 31 (13) ◽  
pp. 2050104
Author(s):  
Shu-Cheng Chang ◽  
Yuxin Dong ◽  
Yingbo Han
Keyword(s):  
Riemannian Manifold ◽  
Heat Flow ◽  
Global Existence ◽  
Sectional Curvature ◽  
Compact Riemannian Manifold ◽  
Sasakian Manifold ◽  
Asymptotic Convergence ◽  
Target Manifold

In this paper, we consider the heat flow for [Formula: see text]-pseudoharmonic maps from a closed Sasakian manifold [Formula: see text] into a compact Riemannian manifold [Formula: see text]. We prove global existence and asymptotic convergence of the solution for the [Formula: see text]-pseudoharmonic map heat flow, provided that the sectional curvature of the target manifold [Formula: see text] is non-positive. Moreover, without the curvature assumption on the target manifold, we obtain global existence and asymptotic convergence of the [Formula: see text]-pseudoharmonic map heat flow as well when its initial [Formula: see text]-energy is sufficiently small.

CLASSES OF GRADIENT RICCI SOLITONS

2011 ◽  
Vol 08 (04) ◽  
pp. 783-796 ◽  
Author(s):  
GABRIEL BERCU ◽  
MIHAI POSTOLACHE
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Wide Class ◽  
Positive Function ◽  
Polar Coordinates ◽  
Ricci Solitons ◽  
Positive Sectional Curvature ◽  
Gradient Ricci Solitons ◽  
Research Article

We introduce a study of Riemannian manifold M = ℝ2 endowed with a metric of diagonal type of the form [Formula: see text], where g is a positive function, of C∞-class, depending on the variable x2 only. We emphasize the role of metric [Formula: see text] in determining manifolds having negative, null or positive sectional curvature. Within this framework, we find a wide class of gradient Ricci solitons (see, Theorems 4 and 7) and specialize these results to discuss some 2D and 4D case studies. The present study can be thought as a natural continuation of those included in monograph [22] by Constantin Udrişte, and to those in the research article [12] by Richard S. Hamilton (the result in Proposition 8 is precisely the famous "Hamilton cigar" in polar coordinates).

scholarly journals Hölder Compactification for Some Manifolds with Pinched Negative Curvature Near Infinity

2008 ◽  
Vol 60 (6) ◽  
pp. 1201-1218 ◽  
Author(s):  
Eric Bahuaud ◽  
Tracey Marsh
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Exponential Map ◽  
Noncompact Riemannian Manifold ◽  
Compact Submanifold ◽  
Curvature Pinching ◽  
Negative Sectional Curvature

AbstractWe consider a complete noncompact Riemannian manifold M and give conditions on a compact submanifold K ⊂ M so that the outward normal exponential map off the boundary of K is a diffeomorphism onto M\K. We use this to compactify M and show that pinched negative sectional curvature outside K implies M has a compactification with a well-defined Hölder structure independent of K. The Hölder constant depends on the ratio of the curvature pinching. This extends and generalizes a 1985 result of Anderson and Schoen.

scholarly journals Geodesic flows on manifolds of negative curvature with smooth horospheric foliations

1991 ◽  
Vol 11 (4) ◽  
pp. 653-686 ◽  
Author(s):  
Renato Feres
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Tangent Bundle ◽  
Geodesic Flow ◽  
Negative Curvature ◽  
Geodesic Flows ◽  
Constant Negative Curvature ◽  
Unit Tangent Bundle ◽  
Tangent Bundles ◽  
Negative Sectional Curvature

AbstractWe improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.

scholarly journals THE SMOOTHNESS OF RIEMANNIAN SUBMERSIONS WITH NON-NEGATIVE SECTIONAL CURVATURE

2005 ◽  
Vol 07 (01) ◽  
pp. 137-144
Author(s):  
JIANGUO CAO ◽  
MEI-CHI SHAW
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Riemannian Submersion ◽  
Fundamental Theory ◽  
Riemannian Submersions ◽  
Flat Strip ◽  
Negative Sectional Curvature

Let Mn be a complete, non-compact and C∞-smooth Riemannian manifold with non-negative sectional curvature. Suppose that [Formula: see text] is a soul of Mn given by the fundamental theory of Cheeger and Gromoll, and suppose that [Formula: see text] is a distance non-increasing retraction from the whole manifold to the soul (e.g. the retraction given by Sharafutdinov). Then we show that the retraction Ψ above must give rise to a C∞-smooth Riemannian submersion from Mn to the soul [Formula: see text]. Moreover, we derive a new flat strip theorem associated with the Cheeger–Gromoll convex exhaustion for the manifold above.

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