We will introduce spaces of riemannian metrics on a smooth manifold satisfying a curvature condition given by a subset in the space of algebraic curvature operators. Provided this condition is surgery stable, which is a notion based on the work of S. Hoelzel guaranteeing the condition can be preserved under surgeries of a certain codimension, we can generalize several theorems from positive...
This talk is a survey on results concerning the space $T^{<0}(M)$, which we call the Teichmüller space of negatively curved Riemannian metrics on $M$. It is defined as the quotient space of the space of all negatively curved Riemannian metrics on $M$ modulo the space of all isotopies of $M$ that are homotopic to the identity. This space was shown to have highly non-trivial homotopy when $M$ is...
Joyce’s orbifold construction and the twisted connected sums by Kovalev and Corti-Haskins-Nordström-Pacini provide many examples of compact Riemannian 7-manifolds with holonomy $G_2$. We would like to use this wealth of examples to guess further properties of $G_2$-manifolds and to find obstructions against holonomy $G_2$, taking into account the underlying topological $G_2$-structures.
The...
For a closed, connected direct product Riemannian manifold $(M,g) = (M_1 \times \ldots \times M_l, g_1 + \ldots + g_l)$ we define its multiconformal class $[\![ g ]\!]$ as the totality $\lbrace f_1^2 g_1 + \ldots + f_l^2 g_l \rbrace$ of all Riemannian metrics obtained from multiplying the metric $g_i$ of each factor by a function $f_i^2:M \rightarrow \mathbb{R}_+$. In this talk we discuss how...
Manifolds all of whose geodesics are closed have been studied a lot, but there are only few examples known. The situation is different if one allows in addition for orbifold singularities. In the talk we discuss such examples and their properties. In particular, we explain rigidity phenomena of the geodesic length spectrum and of metrics with all geodesics closed in dimension two.
We use the $\mathop{Spin}^c$ Dirac operator to generalize a formula of Kreck and Stolz for the s invariant of $S^1$ invariant metrics with positive scalar curvature. We then apply it to show that the moduli spaces of metrics with nonnegative sectional curvature on certain 7-manifolds have infinitely many path components. These include certain positively curved Eschenburg and Aloff-Wallach...
The goal of this talk is to discuss the following result: for a manifold homotopy equivalent to $RP^5$, the moduli space of metrics with non-negative sectional (resp. with positive Ricci) curvature has infinitely many path connected components. The proof involves various elements such as Brieskorn spheres, Grove-Ziller metrics, reduced eta-invariants and fixed point formulas. This is joint...
The classification of positively (sectional) curved manifolds is a long standing open problem in Riemannian geometry. So far it was a successful approach to consider the problem under the extra assumption of an isometric group action.
In this talk I will report on recent joint work with Lee Kennard and Burkhard Wilking in this direction. Among other things we show the following: Let $M$ be a...
In his famous book on partial differential relations Gromov formulates an exercise concerning local deformations of solutions to open partial differential relations. We will explain the content of this fundamental assertion and sketch a proof.
We will illustrate this by various examples, including the construction of $C^{1,1}$-Riemannian metrics which are positively curved "almost everywhere"...
