Riemannian foliations,

and related topics

Jagiellonian University

Kraków, Poland

- Jesús Álvarez López, Santiago de Compostela, Spain; website
- Sigmundur Gudmundsson, Lund, Sweden; website
- Jong Taek Cho, Chonnam National University, South Korea; website
- Georges Habib, Lebanese University, Beirut; website
- Seoung Dal Jung, Jeju National University, South Korea; website
- Felipe Leitner, University of Greifswald, Germany; website
- Thomas Munn, Lund University, Sweden; website
- Liviu Ornea, University of Bucharest, Romania website
- Ken Richardson, Texas Christian University, Fort Worth, Texas, USA; website
- José Ignacio Royo Prieto, Aplicada Ingeniaritza Goi Eskola, UPV-EHU, Bilbao, Spain; website
- Martintxo Saralegi-Aranguren, Université d'Artois, Lens, France; website
- Christina Tonnensen-Friedman, Union Collage, Schenectady, New York, USA; website
- From the Institute of Mathematics of the Jagiellonian University:

**Jesús Álvarez López:***A trace formula for foliated flows*

**Presentation Slides**

Abstract: Let $\mathcal{F}$ be a smooth codimension one foliation on a compact manifold $M$. A flow $\phi^t$ on M is said to be foliated if it maps leaves to leaves. If moreover $\phi^t$ has simple closed orbits and transversely simple preserved leaves, then there are finitely many preserved leaves, which are compact; their union is a compact submanifold $M^0$, and a precise description of the transverse structure of $\mathcal{F}$ can be given. A version of the reduced leafwise cohomology, $\overline{H}I(\mathcal{F})$, is defined by using distributional leafwise differential forms conormal to $M^0$. We define a Lefschetz distribution $L_{\mathrm{dis}}(\phi^t)$ on $\mathbb{R}$ induced by the action ${\phi^t}^*$ on $\overline{H}I(\mathcal{F})$. Then we prove a distributional Lefschetz trace formula describing $L_{\mathrm{dis}}(\phi^t)$ in terms of infinitessimal data of the closed orbits and preserved leaves. This kind of distributional trace formula was conjectured by Christopher Deninger, motivated by possible arithmetic applications. In the case where $M^0=\emptyset$; (when $\mathcal{F}$ is Riemannian), this formula was proved using smooth leafwise differential forms.

**Aleksandra Borówka:***Quaternion-Kähler manifolds near maximal fixed point sets of $S^1$ -symmetries*

**Presentation Slides**

Abstract: In this talk we will present a local characterization of Quaternion-Kähler manifolds near maximal fixed point sets of rotating $S^1$ -symmetries. We apply joint results with D. Calderbank to the case when the resulting quaternionic structure admits a compatible metric: in the original approach, called quaternionic Feix--Kaledin construction, we provide a twistorial construction of a quaternionic structure on a neighbourhood of the zero section of a twisted tangent bundle of a c-projective manifold $S$ of type $(1,1)$. The obtained structure admits a natural $S^1$ action given by the scalar multiplication in the fibres and the zero section is the fixed point set. In the quaternion-Kähler case $S$ must be Kähler but we need to provide a necessary and suffictient conditions on possible twists. It turns out that they are given by a holomorphic line bundle with a connection $(L,\nabla)$ on $S$, where $\nabla$ is a unitary connection with curvature proportional to the Kähler form. In particular, if $S$ is Kähler--Einstein then such twists can be given by (positive and negative) roots of the cannonical bundle. We will start the talk by recalling some basic definitions and giving an overview of the quaternionic Feix--Kaledin construction. Then we will show how the Kähler metric together with a twist satisfying the conditions given above, induce a holomorphic contact structure on the twistor space arising by the quaternionic Feix--Kaledin construction. Finally, we will discuss why the conditions are necessary.

**Jong Taek Cho:***Spherical CR-symmetric hypersurfaces in Hermitian symmetric spaces*

**Presentation Slides**

Abstract: A spherical CR manifold is a contact strongly pseudo-convex CR manifold which is locally CR-equivalent to the sphere endowed with the standard CR-structure as a real hypersurface of complex Euclidean space. In this talk, we give a classfication of spherical CR-symmetric spaces of dimension >3, which are realized as real hypersurfaces in Hermitian symmetric spaces. Also, we classify 3-dimensional contact CR-symmetric spaces.

**Sigmundur Gudmundsson:***Conformal and Minimal Foliations on the Classical Riemannian Symmetric Spaces*

**Presentation Slides**

Abstract: In differential geometry, the study of minimal submanifolds $N$ of a given Riemannian manifold $(M,g)$ is of great importance. In many cases $N$ is a surface of dimension two or a hypersurface of codimension one. In this talk we will discuss the case when the codimension of $N$ is two. It turns out that complex-valued submersive harmonic morphisms $\phi:(M,g)\to{\mathbb C}$ are useful tools for the construction of these beautiful geometric objects. Their fibres are not only minimal but together they form a conformal foliation of codimension two. Such harmonic morphisms are solutions to a non-linear, over-determined system of partial differential equations. They have a non-trivial existence theory and there even exist simple three-dimensional Riemannian Lie groups for which it can be shown that no solutions exist. In this talk, we shall first describe the origins of this problem and then discuss our so called "method of eigenfamilies" that leads to the construction of solutions on all the classical Riemannian symmetric spaces.

**Seoung Dal Jung:***Harmonic maps on Riemannian foliations*

**Presentation Slides**

Abstract: In this talk, we review harmonic maps between Riemannian foliations. There are two kinds of harmonic maps on Riemannian foliations -- that is, transversally harmonic and $(\mathcal{F}.\mathcal{F}')$-harmonic maps. These are generalized to other harmonic maps, such as biharmonic maps, $f$-harmonic maps, and $F$-harmonic maps, etc. We give recent works about these harmonic maps on foliations.

**Liviu Ornea:***Open questions concerning foliations in LCK geometry*

**Presentation Slides**

Abstract: In the first part of this talk, I shall give an account of locally conformally Kähler geometry, recalling the significant subclasses and the state of the art. The I shall describe several open questions which arise naturally in this geometry, related to foliations.

**Paweł Raźny:***A spectral sequence for free isometric Lie algebra actions*

**Presentation Slides**

Abstract: Assume that $(M,g)$ is a compact Riemannian manifold with a free isometric action of a Lie algebra $\mathfrak{g}$. We present a new spectral sequence, arising from the restriction of the standard filtration of the spectral sequence of the foliation $\mathcal{F}_G$ by orbits of the $\mathfrak{g}$-action to a certain subcomplex of the de Rham complex, which connects the basic cohomology of the foliation, the Lie algebra cohomology of $\mathfrak{g}$ and the de Rham cohomology of $M$. The construction is a generalization of the Gysin long exact sequence in Sasakian Geometry and is an extension of our prior work on $\mathcal{K}$-structures.

**Ken Richardson:***Geometric formality and foliations*

**Presentation Slides**

(joint work with G. Habib and R. Wolak)

Abstract: In this expository talk, we give facts about geometric and topological aspects of closed Riemannian manifolds admitting formal metrics, with interpretations from foliation theory. We also introduce a notion of transverse formality of foliations and indicate some results from ongoing research.

**José Ignacio Royo Prieto:***Hard Lefschetz Property for Isometric flows and $S^3$-actions*

(Joint work with M.Saralegi-Aranguren and R.Wolak)

**Presentation Slides**

Abstract: The Hard Lefschetz Property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property). A new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself. Both properties were recently proved to be equivalent in the case of $K$-contact flows. We show that the HLP is naturally defined for the more general category of isometric flows, and for the category of almost-free $S^3$-actions, which generalizes the rich properties of the 3-Sasakian manifolds. We also show that both versions of the HLP (transversal and global) are equivalent for isometric flows and for certain almost-free $S^3$-actions.

**Martintxo Saralegi-Aranguren:***Basic intersection cohomology*

**Presentation Slides**

Abstract: Although the orbit space M/F of a foliated manifold (M,F) can be quite wild from a topological point of view, it can be effectively studied from a cohomological perspective. There is an extensive literature on this topic, especially in the field of Riemannian foliations. These foliations are regular in the sense that all leaves have the same dimension. What happens if this dimension varies? We are in the realm of singular Riemannian foliations, whose paradigmatic example is the foliation arising from an isometric action of a Lie group. This talk presents some of the results obtained, in collaboration with R. Wolak and J.I. Royo Prieto, in this field. We will discuss the main properties of the basic intersection cohomology (BIC) of a singular Riemannian foliation: finiteness, Poincaré duality, minimality, Lefschetz duality, Gysin sequence.

**Christina Tonnesen-Friedman:***Sasakian geometry on sphere bundles*

**Presentation Slides**

Abstract: This talk will be based primarily on past and ongoing work with Charles P. Boyer. The purpose of the presentation is to discuss the Sasakian geometry on odd dimensional sphere bundles over a smooth projective algebraic variety. In particular, we will apply the so-called fiber join construction for K-contact manifolds, introduced by T. Yamazaki around the turn of the century, to the Sasaki case. One of the ongoing goals is to achieve a better understanding of the existence and non-existence of extremal and constant scalar curvature Sasaki metrics on such manifolds. Other goals include gaining a deeper insight into the topology of the manifolds and to generalize Yamazaki's construction to the orbifold case.