On the topology of nonnegatively curved simply connected 4-manifolds with continuous symmetry
type: Item
instance of: scholarly article
title: On the topology of nonnegatively curved simply connected 4-manifolds with continuous symmetry
zbMATH Open document ID: 0824.53036
DOI: 10.1215/S0012-7094-94-07419-X
author: Catherine Searle
author: Da-Gang Yang
published in: Duke Mathematical Journal
publication date: 1995-10-30T00:00:00Z
review text: By discussing a conjecture of \textit{W. Y. Hsiang} and \textit{B. Kleiner} [J. Differ. Geom. 29, No. 3, 615-621 (1989; Zbl 0674.53047)] the authors study the topological classification of some nonnegatively curved 4- dimensional Riemannian manifolds. The following theorem is established: Theorem 1. Let \(M\) be a compact, simply connected, nonnegatively curved 4-dimensional Riemannian manifold with a nontrivial Killing vector field. Then \(M\) is homeomorphic to either \(S^ 4\), \(\mathbb{C} P^ 2\), \(S^ 2\times S^ 2\) or \(\mathbb{C} P^ 2\#\pm\mathbb{C} P^ 2\). If there is an effective isometric \(S^ 1\)-action on \(M\) (\(M\) is called an \(S^ 1\)- Riemannian manifold) such that its fixed point set \(F(S^ 1,M)\) contains two 2-dimensional connected components, then \(M\) is diffeomorphic to \(S^ 2\times S^ 2\) or \(-\mathbb{C} P^ 2\#\mathbb{C} P^ 2\). The second result of the paper is a double soul theorem for a complete nonnegatively curved \(n\)-dimensional \(S^ 1\)-Riemannian manifold.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 624867
zbMATH Keywords: simply connected 4-manifolds
zbMATH Keywords: nonnegative curvature
zbMATH Keywords: infinitesimal isometry
zbMATH Keywords: totally geodesic submanifolds
zbMATH Keywords: isometric \(S^ 1\)-action
reviewed by: I. D. Albu
MaRDI profile type: Publication
cites work: Vector fields and characteristic numbers
cites work: Infinitesimal isometries and Pontryagin numbers
cites work: Some examples of manifolds of nonnegative curvature
cites work: Q4067705
cites work: On the structure of complete manifolds of nonnegative curvature
cites work: An application of gauge theory to four dimensional topology
cites work: Manifolds with positive curvature
cites work: The topology of four-dimensional manifolds
cites work: On complete open manifolds of positive curvature
cites work: Curvature, diameter and Betti numbers
cites work: New Extremal Problems for the Riemannian Recognition Program via Alexandrov Geometry
cites work: Positively curved manifolds with maximal symmetry-rank
cites work: On the topology of positively curved 4-manifolds with symmetry
cites work: Q3730551
cites work: Q5341483
cites work: Riemannian spaces having their curvature bounded below by a positive number
cites work: On the topology of nonnegatively curved simply connected 4-manifolds with discrete symmetry
Recommended article: On the topology of positively curved 4-manifolds with symmetry
Recommended article: On nonnegatively curved 4-manifolds with discrete symmetry
Recommended article: ON POSITIVELY CURVED FOUR-MANIFOLDS WITH S1-SYMMETRY
Recommended article: Almost non-negatively curved 4-manifolds with torus symmetry
Recommended article: Some results on the topology of four-manifolds with nonnegative curvature
Recommended article: On the geography of simply connected nonspin symplectic 4-manifolds with nonnegative signature
Recommended article: On finite symmetries of simply connected four-manifolds
Recommended article: A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry
Recommended article: On simply connected noncomplex 4-manifolds
description: scientific article; zbMATH DE number 624867