My research is based in Riemannian geometry, and the objects of study are Riemannian manifolds with curvature bounds, symmetry, or density. Many of the tools involved in my research are topological (e.g., Steenrod and other cohomology operations, group cohomology, and results from surgery theory), so I am also interested in problems in these areas.
I enjoy collaborating. Here is my list, so far, of coauthors and links to their webpages:
For preprints, please see my arXiv page.
For MathSciNet reviews, my Author ID is 852611.
- L. Kennard, W. Wylie. Positive weighted sectional curvature (preprint). Abstract: In this paper, we give a new generalization of positive sectional curvature called positive weighted sectional curvature. It depends on a choice of Riemannian metric and a smooth vector field. We give several simple examples of Riemannian metrics which do not have positive sectional curvature but support a vector field that gives them positive weighted curvature. On the other hand, we generalize a number of the foundational results for compact manifolds with positive sectional curvature to positive weighted curvature. In particular, we prove generalizations of Weinstein's theorem, O'Neill's formula for submersions, Frankel's theorem, and Wilking's connectedness lemma. As applications of these results, we recover weighted versions of topological classification results of Grove-Searle and Wilking for manifolds of high symmetry rank and positive curvature.
- M. Amann, L. Kennard. On a generalized conjecture of Hopf with symmetry (preprint). Abstract: 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.
- L. Kennard. On the Chern problem with symmetry (preprint). Abstract: In 1965, Chern posed a question concerning the extent to which fundamental groups of positively curved manifolds look like spherical space form groups. Specifically, for a Riemannian manifold with positive sectional curvature, he asked whether every abelian subgroup of the fundamental group cyclic. In 1998, Shankar answered ChernÕs question in the negative. In this article, we consider modifications of ChernÕs question in the presence of symmetry.
- M. Amann, L. Kennard. Positive curvature and rational ellipticity, Algebr. Geom. Topol. (to appear) - - - arXiv. Abstract: Simply-connected manifolds of positive sectional curvature M 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.
- M. Amann, L. Kennard. Topological properties of positively curved manifolds with symmetry, Geom. Funct. Anal. 24 (2014), no. 5, 1377–1405. - - - MathSciNet: MR3261629 Abstract: We obtain upper bounds for the Euler characteristic of a positively curved Riemannian manifold that admits a large isometric torus action. We apply our results to prove obstructions to symmetric spaces, products of manifolds, and connected sums admitting positively curved metrics with symmetry.
- L. Kennard. Positively curved Riemannian metrics with logarithmic symmetry rank bounds, Comment. Math. Helv. (to appear) - - - arXiv. Abstract: We prove an obstruction at the level of rational cohomology to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As one application, we provide evidence for a generalized conjecture of H. Hopf, which states that no symmetric space of rank at least two admits a metric with positive curvature. Other applications concern product manifolds, connected sums, and manifolds with nontrivial fundamental group.
- L. Kennard. On the Hopf conjecture with symmetry, Geom. Topol. 161 (2013), no. 1, 563–593. - - - MathSciNet: MR3039770 (conference proceedings in Lecture Notes in Mathematics, Vol. 2110, with the same title) Abstract: The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new tool is the action of the Steenrod algebra on cohomology.
- J. Holdener, L. Kennard, M. Zaremsky. Generalized Thue-Morse sequences and the von Koch curve. Int. J. Pure Appl. Math. 47 (2008), no. 3, 397–403. --- MathSciNet: MR2458634 Abstract: In a recent paper, Ma and Holdener used turtle geometry and polygon maps to show that the Thue-Morse sequence encodes the von Koch curve. In the final paragraph of this same paper, they ask whether or not there exist certain generalized Thue-Morse sequences that also encode the curve. Here we answer this question in the affirmative, providing an infinite family of words that generate generalized Thue-Morse sequences encoding the von Koch curve.
Work partially supported by NSF Grants DMS-1045292 and DMS-1404670.