Related papers: Symmetries of Eschenburg spaces and the Chern prob…

The present article explores the relationship between positive sectional curvature and the geometric and topological properties of Eschenburg $6$-orbifolds. First, we prove that positive sectional curvature imposes restrictions on the their…

In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar…

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have…

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the…

We investigate the curvature of Eschenburg spaces with respect to two different metrics, one constructed by Eschenburg and the other by Wilking. With respect to the Eschenburg metric, we obtain a simple complete characterization of the…

We examine several classes of manifolds which have the same cohomology ring as an Eschenburg space (a family of biquotients which is a main source of manifolds with positive curvature). One family are the 3-sphere bundles over CP^2. Another…

In this paper we give a classification of cohomogeneity one manifolds admitting an invariant metric with quasipositive sectional curvature except for two $7$-dimensional families. The main result carries over almost verbatim from the…

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there…

In this article we study compact K\ahler manifolds satisfying a certain nonnegativity condition on the bisectional curvature. Under this condition, we show that the scalar curvature is nonnegative and that the first Chern class is positive…

We give an exhaustive description of all simply connected odd dimensional cohomogeneity one manifolds that can possibly support an invariant metric with positive sectional curvature. Among the known examples of odd dimensional manifolds…

We examine the class of compact Hermitian manifolds with constant holomorphic sectional curvature. Such manifolds are conjectured to be K\"ahler (hence a complex space form) when the constant is non-zero and Chern flat (hence a quotient of…

We study isometric circle actions on 7 dimensional positively curved Eschenburg spaces which are almost free, thus giving rise to orbifold fibrations of these spaces. This shows in particular that every known example of compact manifolds…

A central theme in Riemannian geometry is understanding the relationships between the curvature and the topology of a Riemannian manifold. Positive isotropic curvature (PIC) is a natural and much studied curvature condition which includes…

This is a survey on cohomogeneity one manifolds with positive curvature. We discuss the known examples of this type and their geometry and the functions that describe the metric. We also describe the classification of cohomogeneity one…

We show that a compact quaternionic-K\"ahler manifold with positive scalar curvature and nonnegative sectional curvature is isometric to a symmetric space. This extends a classical theorem of Berger.

In each dimension of the form $4n-1$ with $n\geq 3$, we construct infinitely many new examples of manifolds admitting metrics with positive sectional curvature almost everywhere. In addition, we show that if $n\geq 6$, infinitely many of…

We introduce Riemannian metrics of positive scalar curvature on manifolds with Baas-Sullivan singularities, prove a corresponding homology invariance principle and discuss admissible products. Using this theory we construct positive scalar…

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G…

One of the main methods of constructing new spaces with positive or almost positive curvature is the study of biquotients first studied in detail by Eschenburg. We classify orbifold biquotients of the Lie Group $SU(3)$, and construct a new…

Given a non compact semisimple Lie group $G$ we describe all homogeneous spaces $G/L$ carrying an invariant almost K\"ahler structure $(\omega,J)$. When $L$ is abelian and $G$ is of classical type, we classify all such spaces which are…