Abstracts

Speakers 

• Renato Bettiol

• Karen Butt

• Juan Manuel Lorenzo Naveiro

• Xiaochun Rong

• Regina Rotman

• Victor Sanmartin-Lopez

• Catherine Searle

• Ravi Shankar

• Fred Wilhelm

• Matthias Wink

• Kai Xu

Abstracts for the invited talks

• Renato Bettiol - Curvature-homogeneous 4-manifolds

A Riemannian metric is curvature-homogeneous if its curvature operators are the same at all points, using linear isometries between tangent spaces at different points to compare them. Of course, locally homogeneous metrics are curvature-homogeneous, but, in dimensions > 2, these are far from the only examples. In this talk, I will discuss several new results about curvature-homogeneous manifolds of dimension 4, obtained in joint work with Robert Bryant.

• Karen Butt - Marked Poincare rigidity

For closed negatively curved manifolds, the lengths of closed geodesics, marked by their free homotopy classes, are known to characterize the underlying metric up to isometry in many cases. In this talk, we consider a dynamically flavored variant of this marked length spectrum rigidity problem. We introduce the marked Poincare determinant, which associates to each free homotopy class of closed curves a number which measures the unstable volume expansion of the geodesic flow along the associated closed geodesic. Our main result is that near hyperbolic metrics in dimension 3, this invariant determines the metric up to homothety. The key ingredient of the proof is the injectivity of the Lichnerowicz Laplacian on trace-free divergence-free symmetric 2-tensors at a hyperbolic metric in dimension 3. This is joint work with Erchenko, Humbert, Lefeuvre, and Wilkinson. 

• Juan Manuel Lorenzo Naveiro - Nearly Kähler geometry and totally geodesic submanifolds

An almost Hermitian manifold $(M,J)$ is nearly Kähler if the covariant derivative of $J$ is totally skew-symmetric. The study of six dimensional (strictly) nearly Kähler manifolds with nonconstant curvature is particularly interesting, as these are automatically Einstein and, after an appropriate rescaling, their Riemannian cones have holonomy $\mathsf{G}_2$. A theorem of Butruille asserts that the simply connected and homogeneous nearly Kähler $6$-manifolds are the sphere $\mathsf{S}^6$, the complex projective space $\mathbb{C}\mathsf{P}^3$, the flag manifold $\mathsf{F}(\mathbb{C}^3)$ and the almost product $\mathsf{S}^3\times \mathsf{S}^3$.

The aim of this talk is to discuss a joint work with Alberto Rodríguez-Vázquez in which we classify the totally geodesic submanifolds of the aforementioned spaces, as well as their cohomogeneity one $\mathsf{G}_2$-cones. In order to do this, we will introduce the necessary techniques to study totally geodesic submanifolds in homogeneous spaces, as well as on Riemannian cones.

• Xiaochun Rong - TBD

• Regina Rotman - TBD

• Victor Sanmartin-Lopez - Curvature-adaptedness and cohomogeneity one actions in symmetric spaces

In submanifold geometry, it is natural to begin by investigating these submanifolds with a high degree of symmetry, such as homogeneous hypersurfaces, or, equivalently, cohomogeneity one actions. Indeed, one of the main goals of this talk is to present the classification of cohomogeneity one actions on symmetric spaces of non-compact type.

We will also provide some details concerning the extrinsic geometry of the examples of the classification. More precisely, we will focus on the ones that are curvature-adapted or minimal.

• Catherine Searle - How to lift Positive Intermediate Ricci Curvature

The pth intermediate Ricci curvature interpolates between Ricci curvature and sectional curvature. In previous work of Searle and Wilhelm, they gave conditions under which it is possible to lift positive Ricci curvature from the quotient space of a G-manifold to the manifold itself, while simultaneously observing that positive sectional curvature cannot be lifted. It is then of interest to understand for which p we can lift positive pth intermediate Ricci curvature. Using recent work of Reiser and Wraith establishing conditions under which the total space of a fibration with totally geodesic fibers admits positive pth intermediate Ricci curvature, combined with techniques from the work of Searle and Wilhelm, we prove a lifting theorem for G-manifolds of positive pth intermediate Ricci curvature.

• Ravi Shankar -  Non-negative curvature metrics on vector bundles over 7-spheres

 We present recent results on the construction of (infinitely many) metrics of non-negative curvature on every vector bundle over each homotopy 7-sphere. This answers the question of the existence of such metrics as part of the Inverse Soul Problem of Cheeger and Gromoll.  This is joint work with David Duncan and Rebecca Field.

• Fred Wilhelm - Walschap meets Gauss and Cheeger 

• Matthias Wink - Curvature and cohomology of Kaehler manifolds

A celebrated result of Mori and Sui-Yau says that manifolds with positive bisectional curvature are biholomorphic to complex projective space. In this talk we will introduce new curvature conditions that provide characterizations of cohomology complex projective spaces. For example, the curvature tensor of a Kaehler manifold induces an operator on symmetric holomorphic 2-tensors, called Calabi operator. This operator is the identity for complex projective space with the Fubini Study metric. We show that a compact n-dimensional Kaehler manifold with n/2-positive Calabi curvature operator has the rational cohomology of complex projective space. The complex quadric shows that this result is sharp if n is even. This talk is based on joint work with K. Broder, J. Nienhaus, P. Petersen, J. Stanfield.

• Kai Xu - TBD