As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of S × S....

Let N be the class of closed simply connected, smooth n–manifolds that admit nonnegative sectional curvature and let P be the corresponding class for positive curvature. Known examples suggest that N ought to be much larger than P . On the other hand, there is no known obstruction that distinguishes between the two classes. So its actually possible that N = P . In [PetWilh2] we will give a defo...

One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being...

For k ≥ 2, let M4k−1 be a closed (2k−2)-connected manifold. If k ≡ 1 mod 4 assume further that M is (2k−1)-parallelisable. Then there is a homotopy sphere Σ4k−1 such that M]Σ admits a Ricci positive metric. This follows from a new description of these manifolds as the boundaries of explicit plumbings.

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain a point at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces together with quotients of S×S which admit metrics with this property. It is an unfortunate fact tha...

We show that a certain class of manifolds admit metrics of positive Ricci curvature This class includes many exotic spheres including all homotopy spheres which represent elements of bP n

A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riem...

We prove some results about the vanishing of the elliptic genus on positively curved Spin manifolds with logarithmic symmetry rank. The proofs are based on the rigidity of the elliptic genus and Kennard’s improvement of the Connectedness Lemma for transversely intersecting, totally geodesic submanifolds.