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 Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

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.

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.

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.

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.

TQFT and Whitehead's Manifold

1997 ◽  
Vol 06 (01) ◽  
pp. 13-30
Author(s):  
Louis Funar
Keyword(s):  
Field Theories ◽  
Quantum Field Theories ◽  
Quantum Field ◽  
Open Manifold ◽  
Open Manifolds ◽  
Topological Quantum ◽  
Level 3 ◽  
Simply Connected ◽  
Topological Quantum Field Theories

The aim of this note is to give two applications of Topological Quantum Field Theories to the topology of open manifolds. The invariants we derived may be used to test if an open manifold is simply connected at infinity (as we did for Whitehead's manifold in case of the sl2(C)-TQFT in level 3) or to construct examples of non-homeomorphic contractible open 3-manifolds.

scholarly journals A general rigidity theorem for complete submanifolds

1998 ◽  
Vol 150 ◽  
pp. 105-134 ◽  
Author(s):  
Katsuhiro Shiohama ◽  
Hongwei Xu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Second Fundamental Form ◽  
Geometric Inequalities ◽  
Positive Sectional Curvature ◽  
The Second Fundamental Form ◽  
Complete Submanifolds ◽  
Parallel Mean Curvature ◽  
Rigidity Property ◽  
Simply Connected

Abstract.Making use of 1-forms and geometric inequalities we prove the rigidity property of complete submanifolds Mn with parallel mean curvature normal in a complete and simply connected Riemannian (n+p) -manifold Nn+p with positive sectional curvature. For given integers n, p and for a nonnegative constant H we find a positive number T(n,p) ∈ (0,1) with the property that if the sectional curvature of N is pinched in [T(n,p), 1], and if the squared norm of the second fundamental form is in a certain interval, then Nn+p is isometric to the standard unit (n + p)-sphere. As a consequence, such an M is congruent to one of the five models as seen in our Main Theorem.

scholarly journals ON SOME HYPERCOMPLEX 4-DIMENSIONAL LIE GROUPS OF CONSTANT SCALAR CURVATURE

2009 ◽  
Vol 06 (04) ◽  
pp. 619-624 ◽  
Author(s):  
HAMID REZA SALIMI MOGHADDAM
Keyword(s):  
Scalar Curvature ◽  
Sectional Curvature ◽  
Lie Groups ◽  
Constant Scalar Curvature ◽  
Positive Sectional Curvature ◽  
Explicit Formulas ◽  
Hermitian Metrics ◽  
Hypercomplex Structure ◽  
Sectional Curvatures ◽  
Simply Connected

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi–Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.

scholarly journals Normal subgroups of diffeomorphism and homeomorphism groups of ℝn and other open manifolds

2011 ◽  
Vol 31 (6) ◽  
pp. 1835-1847 ◽  
Author(s):  
PAUL A. SCHWEITZER, S. J.
Keyword(s):  
Normal Subgroup ◽  
Compact Manifold ◽  
Normal Subgroups ◽  
Open Manifold ◽  
Open Manifolds ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

AbstractWe determine all the normal subgroups of the group of Cr diffeomorphisms of ℝn, 1≤r≤∞, except when r=n+1 or n=4, and also of the group of homeomorphisms of ℝn ( r=0). We also study the group A0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with non-empty boundary, then the quotient of A0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.

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