Kristopher Tapp

Invariant Metrics with Nonnegative Curvature on Compact Lie Groups

We classify the left-invariant metrics with nonnegative sectional curvature on SO(3) and U(2).

Conditions for nonnegative curvature on vector bundles and sphere bundles

This paper addresses Cheeger and Gromoll’s question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle which admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize Walschap and Strake’s conditions under which a vector bundle admits a connection metric of nonnegative curvature.

Finiteness theorems for submersions and souls

The first section of this paper provides an improvement upon known finiteness theorems for Riemannian submersions; that is, theorems which conclude that there are only finitely many isomorphism types of fiber bundles among Riemannian submersions whose total spaces and base spaces both satisfy certain geometric bounds. The second section of this paper provides a sharpening of some recent theorems which conclude that, for an open manifold of nonnegative curvature satisfying certain geometric bounds near its soul, there are only finitely many possibilities for the isomorphism class of a normal bundle of the soul. A common theme to both sections is a reliance on basic facts about Riemannian submersions whose A and T tensors are both bounded in norm.

Nonnegatively Curved Metrics on S2 × ℝ2

We classify the complete metrics of nonnegative sectional curvature on M2 × ℝ2, where M2 is any compact 2-manifold.

Rigidity for Nonnegatively Curved Metrics on S2 × R3

We address the question: how large is the family of complete metricswith nonnegative sectional curvature on S2 × R3? We classify theconnection metrics, and give several examples of nonconnection metrics.We provide evidence that the family is small by proving some rigidityresults for metrics more general than connection metrics.

Volume growth and holonomy in nonnegative curvature

The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between the codimension of the soul and the maximal dimension of an orbit of the action of the normal holonomy group of the soul. Additionally, an example of a simplyconnected soul with a non-compact normal holonomy group is constructed.

Cohomogeneity one disk bundles with normal homogeneous collars

We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundle, then we show that its rank has to be in

Nonnegatively and positively curved invariant metrics on circle bundles

\{2,3,4,6,8\}

The Mathematics of Measuring Self-Delusion

. Moreover, we give a complete classification of such bundles of rank 6 and 8, and a partial classification for rank 3.

The Five Platonic Solids

We derive and study necessary and sufficient conditions for an S 1 -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial .bundle S 1 x S 3 that is not a product metric.