On the class of metrics g on 
                $${\mathbb {S}}^2\times {\mathbb {S}}^2$$
                
                    
                        
                            
                                
                                    S
                                
                                2
                            
                            ×
                            
                                
                                    S
                                
                                2
                            
                        
                    
                
             conformal to the standard metric 
                $$g_0$$
                
                    
                        
                            g
                            0
                        
                    
                
             induced by 
                $${\mathbb {R}}^6$$
                
                    
                        
                            
                                R
                            
                            6
                        
                    
                
             and the Hopf conjecture

Athoumane Niang; Adama Thiandoum

doi:10.1007/s13370-017-0523-7

On the class of metrics g on $${\mathbb {S}}^2\times {\mathbb {S}}^2$$ S 2 × S 2 conformal to the standard metric $$g_0$$ g 0 induced by $${\mathbb {R}}^6$$ R 6 and the Hopf conjecture

Afrika Matematika ◽

10.1007/s13370-017-0523-7 ◽

2017 ◽

Vol 29 (1-2) ◽

pp. 29-32

Author(s):

Athoumane Niang ◽

Adama Thiandoum

Keyword(s):

Hopf Conjecture

Download Full-text

Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0055 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Yongqiang Liu ◽

Laurenţiu Maxim ◽

Botong Wang

Keyword(s):

Integral Homology ◽

Algebraic Varieties ◽

Finiteness Property ◽

Rational Homology ◽

Mellin Transformation ◽

Hopf Conjecture ◽

Ball Quotients ◽

Proper Map ◽

Complex Projective ◽

Aspherical Manifolds

Abstract In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

Download Full-text

The Hopf conjecture concerning surfaces in 𝐸³

Proceedings of Symposia in Pure Mathematics - Differential Geometry ◽

10.1090/pspum/027.1/9928 ◽

1975 ◽

pp. 135-135

Author(s):

S. I. Goldberg

Keyword(s):

Hopf Conjecture

Download Full-text

On Geroch's Counterexample to the Algebraic Hopf Conjecture

Proceedings of the American Mathematical Society ◽

10.2307/2041496 ◽

1976 ◽

Vol 59 (2) ◽

pp. 334 ◽

Cited By ~ 1

Author(s):

Paul F. Klembeck

Keyword(s):

Hopf Conjecture

Download Full-text

Homogeneous Finsler spaces and the flag-wise positively curved condition

Forum Mathematicum ◽

10.1515/forum-2018-0130 ◽

2018 ◽

Vol 30 (6) ◽

pp. 1521-1537

Author(s):

Ming Xu ◽

Shaoqiang Deng

Keyword(s):

Lie Algebra ◽

Tangent Plane ◽

Finsler Metric ◽

Finsler Metrics ◽

Finsler Spaces ◽

Negatively Curved ◽

Hopf Conjecture ◽

Coset Spaces ◽

Positive Flag Curvature ◽

Positively Curved

Abstract In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space {S^{3}\times\mathbb{R}} which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on {S^{2}\times S^{3}} and {S^{6}\times S^{7}} which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on {S^{3}\times S^{3}} , shedding some light on the long standing general Hopf conjecture.

Download Full-text

The Hopf conjecture for positively curved manifolds with discrete abelian group actions

Differential Geometry and its Applications ◽

10.1016/j.difgeo.2007.11.023 ◽

2008 ◽

Vol 26 (3) ◽

pp. 313-322 ◽

Cited By ~ 2

Author(s):

Xiaole Su ◽

Yusheng Wang

Keyword(s):

Abelian Group ◽

Group Actions ◽

Curved Manifolds ◽

Hopf Conjecture ◽

Abelian Group Actions ◽

Positively Curved ◽

Discrete Abelian Group

Download Full-text

The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-01-06039-7 ◽

2001 ◽

Vol 130 (1) ◽

pp. 163-166 ◽

Cited By ~ 8

Author(s):

Thomas Püttmann ◽

Catherine Searle

Keyword(s):

High Symmetry ◽

Hopf Conjecture ◽

Symmetry Rank

Download Full-text

THE HOPF CONJECTURE FOR MANIFOLDS WITH ABELIAN GROUP ACTIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199705001660 ◽

2005 ◽

Vol 07 (01) ◽

pp. 121-136 ◽

Cited By ~ 6

Author(s):

XIAOCHUN RONG ◽

XIAOLE SU

Keyword(s):

Fixed Point ◽

Abelian Group ◽

Sectional Curvature ◽

Euler Characteristic ◽

Group Actions ◽

Closed Manifold ◽

Positive Sectional Curvature ◽

Hopf Conjecture ◽

Abelian Group Actions

Let M be a closed even n-manifold of positive sectional curvature on which a torus Tk acts isometrically. We show that if [Formula: see text] (respectively, k > 1) for n ≠ 12 (respectively, n = 12), then the Euler characteristic of each Tk-fixed point component is positive. This implies that the Euler characteristic of M is positive. We also extend this result to an isometric elementary p-group [Formula: see text]-action on a closed manifold of positive sectional curvature.

Download Full-text