The even dimensional pinching problem and SU(3)/T
type: Item
instance of: scholarly article
title: The even dimensional pinching problem and SU(3)/T
zbMATH Open document ID: 0669.53035
DOI: 10.1007/BF00572449
published in: Geometriae Dedicata
publication date: 1989-01-01T00:00:00Z
review text: Let M be a compact and connected Riemannian manifold. This manifold M is called \(\delta\)-pinched if its sectional curvature \(\sigma\) (\(\lambda)\) satisfies the inequalities \(0<\delta \leq \sigma (\lambda)\leq 1\) for every plane \(\lambda \in T_ p(M)\) and for every \(p\in M\). The simply connected homogeneous manifolds which admit (homogeneous) metrics with positive sectional curvature have been classified by M. Berger, N. Wallach and P. Berard-Bergery. Aside from the rank 1 symmetric spaces, there are two normal homogeneous spaces \(M^ 7=Sp(2)/SU(2),\) and \(M^{13}=SU(5)/(Sp(2)\times S^ 1),\) three even-dimensional manifolds, \(M^ 6=SU(3)/T,\quad M^{12}=Sp(3)/SU(2)\times SU(2)\times SU(2),\) \(M^{24}=F_ 4/Spin(8)\) and an infinite family of seven manifolds \(SU(3)/S^ 1_{k,\ell}\). Lower bounds have been computed for the above- mentioned odd-dimensional examples, which are 1/37, 16/29 \(\cdot 37\) and 16/29 \(\cdot 37\) for the manifolds Sp(2)/SU(2), \(SU(5)/(Sp(2)\times S^ 1)\) and \(SU(3)/S^ 1_{k,\ell}\) respectively. The main result of this paper can be stated as follows. ``Any homogeneous \(\delta\)-pinched metric on SU(3)/T satisfies \(\delta\leq 4/21''\).
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C30
zbMATH DE Number: 4096308
zbMATH Keywords: sectional curvature
zbMATH Keywords: homogeneous manifolds
zbMATH Keywords: pinched metric
author: Karsten Grove
MaRDI profile type: Publication
cites work: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures
cites work: Les variétés riemanniennes homogenes simplement connexes de dimension impaire à courbure strictement positive
cites work: Q3283409
cites work: Sur les variétés riemanniennes pincées juste au-dessous de 1/4
cites work: Finiteness Theorems for Riemannian Manifolds
cites work: Q4067705
cites work: Die Krümmung des Raumes \(Sp(2)/SU(2)\) von Berger
cites work: New examples of manifolds with strictly positive curvature
cites work: Q3344797
cites work: Lipschitz convergence of Riemannian manifolds
cites work: Q5683423
cites work: Group actions and curvature
cites work: The curvature of SU (5)/(\(Sp(2) \times S^1\))
cites work: Q3935090
cites work: Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds.
cites work: Q4727799
cites work: Pinching theorem for the real projective space
cites work: Q3759510
cites work: Compact homogeneous Riemannian manifolds with strictly positive curvature
full work available at URL: https://doi.org/10.1007/bf00572449
OpenAlex ID: W1972427397
Recommended article: The Pinch Technique and its Applications to Non-Abelian Gauge Theories
Recommended article: Confinement, screening and the center on \(S^{3} \times S^{1}\)
Recommended article: Conjugate connections and SU(3)-instanton invariants
Recommended article: A perturbative \(SU(3)\) Casson invariant
Recommended article: Q4546994
Recommended article: Solution of \(N=2\) gauge theories via compactification to three dimensions.
Recommended article: Mixed type solutions of the \(\mathrm{SU}(3)\) models on a torus
Recommended article: SU(3) Chern-Simons field theory in three-manifolds
Recommended article: SUSY gauge theories on squashed three-spheres
description: scientific article; zbMATH DE number 4096308