# Riemannian Topology Meeting

Europe/Berlin
University of Fribourg, Switzerland

Description

## Augsburg-Fribourg-Karlsruhe RTM 2018

Spaces of riemannian metrics and related topics

The Riemannian Topology Meeting (RTM) is an irregularly recurring activity addressing mathematicians interested in the interplay of algebraic and geometric topology with riemannian geometry. Its tradition reaches back to 2012 during which it took place at the universites of Fribourg, Karlsruhe and Göttingen.

This RTM, taking place at the University of Fribourg, focuses on fostering the mathematical intercommunication between the research groups in Augsburg, Fribourg and Karlsruhe, which are participating in the SPP 2026 DFG project.

## Speakers

• Johannes Ebert (WWU Münster)
• Sebastian Goette (University of Freiburg)
• Jackson Goodman (University of Pennsylvania)
• Christian Lange (University of Cologne)
• Saskia Roos (University of Potsdam)
• Gangotryi Sorcar (Einstein Institute of Mathematics, HUJI)
• Michael Wiemeler (WWU Münster)
• David Wraith (Maynooth University)

There will be three additional talks each given by a member of the groups in Augsburg, Fribourg and Karlsruhe, respectively.

## Supported by

• Anand Dessai (University of Fribourg)
• Bernhard Hanke (Augsburg University)
• Wilderich Tuschmann (Karlsruhe Institute of Technology)
Participants
• Ana Karla Garcia Perez
• Anand Dessai
• Benedikt Hunger
• Bernhard Hanke
• Christian Lange
• Daniel Räde
• David Degen
• David González
• David Wraith
• Diego Corro Tapia
• Eric Schlarmann
• Fernando Galaz-Garcia
• Gangotryi Sorcar
• Georg Frenck
• Hemanth Saratchandran
• Ivan Izmestiev
• Jackson Goodman
• Jan-Bernhard Kordaß
• Johannes Ebert
• Jonathan Wermelinger
• Kevin Kalbermatter
• Llohann Dallagnol Sperança
• Manuel Amann
• Martin Günther
• Martin Kerin
• Masoumeh Zarei
• Mauricio Bustamante
• Michael Wiemeler
• Moritz Meisel
• Patrick Ghanaat
• Philipp Reiser
• Sebastian Goette
• Stephan Klaus
• Séverine Oppliger
• Thorsten Hertl
• Wilderich Tuschmann
• Wolfgang Steimle
• Thursday, 8 November
• 08:45 09:10
Registration 25m
• 09:10 09:20
Opening 10m
• 09:20 10:10
A rigidity theorem for the action of the diffeomorphism group on spaces of psc metrics 50m

For a closed, simply connected $d$-dimensional manifold spin $M$, we study the action of the (spin) diffeomorphism group of $M$ on the space $\mathcal{R}^+ (M)$ of psc metrics on $M$. Our main result is that the homotopy class of the map $f^*: \mathcal{R}^+ (M) \to \mathcal{R}^+ (M)$ only depends on the cobordism class in $\Omega^{\mathrm{Spin}}_{d+1}$ of the mapping torus of $f$. When properly formulated, the same result is true for manifolds with nontrivial fundamental group.

Speaker: Johannes Ebert
• 10:10 10:40
Coffee break 30m Mensa ()

### Mensa

• 10:40 11:30
Spaces of riemannian metrics satisfying surgery stable curvature conditions 50m

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 scalar curvature geometry to this setting. Notably, we will comment on a generalization of a theorem of V. Chernysh on the homotopy type of the space of psc metrics and point to cases where we can distinguish connected components using invariants from spin geometry.

Speaker: Jan-Bernhard Kordaß (Karlsruhe Institute of Technology (KIT))
• 11:30 11:35
Short break 5m
• 11:35 12:25
On the topology of the Teichmüller space of negatively curved Riemannian metrics 50m

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 real hyperbolic by Tom Farrell and Pedro Ontaneda in 2009. Then in 2015, it was shown to be non simply connected in my thesis when $M$ is a suitably chosen Gromov-Thurston manifold (which are examples of negatively curved non-locally symmetric spaces). In 2017, Tom Farrell and myself proved a similar result for $M$ being a suitable complex hyperbolic manifold. In all these results, the dimension of $M$ has to be $4k-2$ for some $k \ge 2$. In this talk, I will explain this project, and talk about the tools we have used so far in unraveling it. I will also mention the cases that are still open in this project.

Speaker: Gangotryi Sorcar (Einstein Institute of Mathematics, HUJI)
• 12:25 14:10
Lunch break 1h 45m Mensa ()

### Mensa

• 14:10 15:00
Extra twisted connected sums and their $\nu$-invariants 50m

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 Crowley-Nordström $\nu$-invariant distinguishes topological $G_2$-structures. It vanishes for all twisted connected sums. By adding an extra twist to this construction, we show that the $\nu$-invariant can assume all of its 48 possible values. This shows that $G_2$-bordism presents no obstruction against holonomy $G_2$. We also exhibit examples of 7-manifolds with disconnected $G_2$-moduli space. Our computation of the $\nu$-invariants involves integration of the Bismuth-Cheeger $\eta$-forms for families of tori, which can be done either by elementary hyperbolic geometry, or using modular properties of the Dedekind $\eta$-function.

Speaker: Sebastian Goette (University of Freiburg)
• 15:00 15:05
Short break 5m
• 15:05 15:55
Scalar curvature and the multiconformal class of a direct product Riemannian manifold 50m

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 constant scalar curvature metrics in a multiconformal class are related with constant scalar curvature metrics on the factors.

Speaker: Saskia Roos (University of Potsdam)
• 15:55 16:30
Coffee break 35m Mensa ()

### Mensa

• 16:30 17:20
Orbifolds all of whose geodesics are closed 50m

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.

Speaker: Christian Lange (University of Cologne)
• 19:30 21:30
Conference Dinner 2h
• Friday, 9 November
• 09:20 10:10
Non-negative Ricci curvature and harmonic maps 50m

Taking as our starting point the classic paper of Eells and
Sampson, we use harmonic maps as a tool to investigate spaces and moduli spaces of Ricci non-negative metrics, and also to study concordances between such metrics. In the first case, among other things, we recover some recent results of Tuschmann and Wiemeler. In the second case, we uncover an interrelationship between concordance, isotopy and isometry for Ricci non-negative metrics, which stands in contrast to the situation for positive scalar curvature.

Speaker: David Wraith
• 10:10 10:40
Coffee break 30m Mensa ()

### Mensa

• 10:40 11:30
$\mathop{Spin}^c$ Dirac operators and the Kreck-Stolz s invariant. 50m

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 spaces. Furthermore, we use a $\mathop{Spin}^c$ version of the s invariant to discuss moduli spaces of metrics of positive scalar and twisted scalar curvature on $\mathop{Spin}^c$ manifolds.

Speaker: Jackson Goodman (University of Pennsylvania)
• 11:30 11:40
Short break 10m
• 11:40 12:30
Moduli spaces of non-negatively curved metrics on homotopy $RP^5$s 50m

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 work with Anand Dessai.​

Speaker: David González Álvaro (University of Fribourg)
• 12:30 14:30
Lunch break 2h Mensa ()

### Mensa

• 14:30 15:20
Positively curved manifolds with isometric torus actions 50m

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 simply connected positively curved $n$-dimensional manifold with $H^{odd} M,\mathbb{Q})=0$ and an isometric $T^8$-action. Then the rational cohomology ring of $M$ is isomorphic to the rational cohomology of one of the CROSSes $S^n$, $\mathbb{C} P^{n/2}$ and $\mathbb{H} P^{n/4}$.

Speaker: Michael Wiemeler (WWU Münster)
• 15:20 15:30
Short break 10m
• 15:30 16:20
Local flexibility of open partial differential relations 50m

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" on arbitrary manifolds.

This is joint work with Christian Bär (Potsdam).

Speaker: Bernhard Hanke (Augsburg University)
• 16:20 16:50
Break 30m Lecture Hall ()

### Lecture Hall

• 16:50 17:40
Discussion 50m
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