Lee Kennard

On the Hopf Conjecture with Symmetry

The Hopf conjecture states that an even-dimensional manifold with positive curvature has positive Euler characteristic. We show that this is true under the assumption that a torus of sufficiently large dimension acts by isometries. This improves previous results by replacing linear bounds by a logarithmic bound. The new method that is introduced is the use of Steenrod squares combined with geometric arguments of a similar type to what was done before.

On a generalized conjecture of Hopf with symmetry

A famous conjecture of Hopf is that the product of the two-dimensional sphere with itself does not admit a Riemannian metric with positive sectional curvature. More generally, one may conjecture that this holds for any nontrivial product. We provide evidence for this generalized conjecture in the presence of symmetry.

Positive curvature and rational ellipticity

Simply-connected manifolds of positive sectional curvature

The Weighted Connection and Sectional Curvature for Manifolds With Density

M

Geometry of manifolds with non-negative sectional curvature

are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, i.e., all but finitely many homotopy groups are conjectured to be finite. In this article we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include a small upper bound on the Euler characteristic and confirmations of famous conjectures by Hopf and Halperin under additional torus symmetry. We prove several cases (including all known even-dimensional examples of positively curved manifolds) of a conjecture by Wilhelm.

On dimensions supporting a rational projective plane

In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.

Characterizations of the round two-dimensional sphere in terms of closed geodesics

Riemannian manifolds with positive sectional curvature.- An introduction to isometric group actions.- A note on maximal symmetry rank, quasipositive curvature and low dimensional manifolds.- Lectures on n-Sasakian manifolds.- On the Hopf conjecture with symmetry.- An Introduction to Exterior Differential Systems.

Topological properties of positively curved manifolds with symmetry

A rational projective plane (

Positively curved Riemannian metrics with logarithmic symmetry rank bounds

\mathbb{QP}^2

Positive weighted sectional curvature

) is a simply connected, smooth, closed manifold