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Events
Transformation Groups in Pseudo-Riemannian Geometry

MPI für Mathematik in den Naturwissenschaften Leipzig

A3 01 (Sophus-Lie room)

conference

29/06/2006 01/07/2006

Transformation Groups in Pseudo-Riemannian Geometry

Program

09:00 - 09:50	Dmitri Alekseevsky (University of Hull, Hull, United Kingdom) Einstein symmetric spaces We discuss the structure of pseudo-Riemannian Einstein symmetric spaces. In particular, we present a classification of quaternionic Kaehler and para-quaternionic Kaehler symmetric spaces with non-zero scalar curvature. We describe also some class of Ricci-flat Kaehler symmetric spaces.
10:00 - 10:50	Jost Eschenburg (Universität Augsburg, Augsburg, Germany) Semi-Riemannian extrinsic symmetric spaces
10:50 - 11:30	Coffee break
11:30 - 12:20	Sergio Console (Università di Torino, Torino, Italy) Submanifolds and holonomy (I) In the present talk I survey applications of holonomic methods to the study of submanifold geometry, showing the consequences of some sort of extrinsic version of de Rham decomposition and Berger's Theorem, the so-called Normal Holonomy Theorem.
12:20 - 15:00	Lunch break
15:00 - 15:50	Carlos Olmos (Universidad Nacional de Córdoba, Córdoba, Argentina) Submanifold and holonomy (II) After surveying, as an introduction, some results related to submanifolds and holonomy we will speak about some joint work with Sergio Console and Antonio Di Scala. Namely, we proved a Berger type theorem for the normal holonomy of complex submanifolds of the complex projective space (also for the complex Euclidean space). Namely, for a full and complete complex projective submanifold M, are equivalent: (i) The normal holonomy is not transitive (i.e. different from U(n), since it is an s-representation). (ii) M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible hermitian symmetric space. (iii) M is extrinsic symmetric (last two equivalences are well known). The methods in the proof rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space) and complex geometry.
15:50 - 16:20	Coffee break
16:20 - 17:10	Antonio José Di Scala (Politecnico di Torino, Torino, Italy) The normal holonomy group of submanifolds with parallel second fundamental form
17:20 - 18:10	Martin Olbrich (Georg-August Universität Göttingen, Göttingen, Germany) Indefinite hyper-Kähler symmetric spaces
19:00 - 20:00	Dinner (on the basis of à la carte and individual payment)

09:00 - 09:50	Shirley Bromberg (Universidad Autonoma Metropolitana, Iztapalapa, Iztapalapa, Mexico) Left-invariant pseudo-Riemannian metrics on Lie groups (I)
10:00 - 10:50	Alberto Medina (Université Montpellier 2, Montpelliert, France) Left-invariant pseudo-Riemannian metrics on Lie groups (II)
10:50 - 11:30	Coffee break
11:30 - 12:20	Michel Goze (Université de Haute Alsace, Mulhouse, France) $\Gamma$-symmetric homogeneous spaces and Lie algebras graded by groups
12:20 - 15:00	Lunch break
15:00 - 15:50	Michel Cahen (Université Libre de Bruxelles, Belgium) Quantizing symmetric spaces
15:50 - 16:20	Coffee break
16:20 - 17:10	José Miguel Figueroa-O'Farrill (University of Edinburgh, United Kingdom) String backgrounds from Lie groups: beyond the WZW model I will explain the role of Lie groups in the construction of (exact) string backgrounds. I will review the appearance of Lie groups admitting a bi-invariant metric in the so-called Wess-Zumino-Witten model and will then report on some work in progress with Noureddine Mohammedi in Tours on constructing exact string backgrounds out of left-invariant metrics on Lie groups.
17:20 - 18:10	Said Benayadi (Université de Metz, Metz, France) Metric Lie superalgebras with reductive even parts We study the structures of (even or odd)-metric (or quadratic) Lie superalgebras over an algebraically closed field of null characteristic. In particular, we characterize the socles and we give inductives descriptions of these Lie superalgebras.

09:00 - 09:50	Ernst Heintze (Universität Augsburg, Augsburg, Germany) Infinite dimensional symmetric spaces of Kac-Moody type Quotients of a Kac-Moody group by the fixed point set of an involution are in some sense the closest generalization of Riemannian symmetric spaces to infinite dimensions. These spaces carry a natural metric of Lorentz type but otherwise share many properties with their finite dimensional counter parts.
10:00 - 10:50	Abdelghani Zeghib (École Normale Supérieure de Lyon, Lyon, France) On the conformal group of Finsler manifolds We generalize to the Finsler case, the Lelong-Ferrand-Obatta Theorem about the compactness of conformal groups of compact Riemannian manifolds, except, the standard sphere.
10:50 - 11:30	Coffee break
11:30 - 12:20	Lionel Berard-Bergery (Université Henri Poincaré Nancy 1, Nancy, France) Invariant subspaces for the holonomy of pseudo-riemannian manifolds and applications
12:20 - 14:30	Lunch break
14:30 - 15:20	Vicente Cortés Suárez (Universität Hamburg, Hamburg, Germany) The twistor spaces of a para-quaternionic Kähler manifold We develop the twistor theory of $G$-structures for which the (linear) Lie algebra of the structure group contains an involution, instead of a complex structure. The twistor space $Z$ of such a $G$-structure is endowed with a field of involutions ${\cal J}\in\Gamma ({\rm End}\, TZ)$ and a ${\cal J}$-invariant distribution ${\cal H}_Z$. We study the conditions for the integrability of ${\cal J}$ and for the (para-)holomorphicity of ${\cal H}_Z$. Then we apply this theory to para-quaternionic Kähler manifolds of non-zero scalar curvature, which admit two natural twistor spaces $(Z^\epsilon, {\cal J}, {\cal H}_Z)$, $\epsilon=\pm 1$, such that ${\cal J}^2=\epsilon$ Id. We prove that in both cases ${\cal J}$ is integrable (recovering results of Blair, Davidov and Muŝkarov) and that ${\cal H}_Z$ defines a holomorphic ($\epsilon=-1$) or para-holomorphic ($\epsilon=+1$) contact structure. Furthermore, we determine all the solutions of the Einstein equation for the canonical one-parameter family of pseudo-Riemannian metrics on $Z^\epsilon$. In particular, we find that there is a unique Kähler-Einstein ($\epsilon=-1$) or para-Kähler-Einstein ($\epsilon=+1$) metric. Finally, we prove that any Kähler or para-Kähler submanifold of a para-quaternionic K\"ahler manifold is minimal and describe all such submanifolds in terms of complex ($\epsilon=-1$), respectively, para-complex ($\epsilon=+1$) submanifolds of $Z^\epsilon$ tangent to the contact distribution. (This is joint work with Dmitri Alekseevsky.)
15:20 - 15:50	Coffee break
15:50 - 16:40	Charles Boubel (École normale supérieure de Lyon, Lyon, France) Lorentzian flows on compact 3-manifolds We call here briefly "flow" a 1-dimensional foliation. In other words, we consider the orbits of nonsingular flows, i.e. the integral curves of nonsingular vector fields on some manifold M, regardless of their parametrization. Besides, let us recall that a (pseudo)-Riewmannian metric transverse to some foliation F on a manifold M is a field of nondegenerate symmetric bilinear forms on the normal bundle n(F)=TM/TF of the foliation F, which is invariant by the flow of any vector field tangent to F. If F is the trivial 0-dimensional foliation by points of M, this gives back the usual definition of a (pseudo)-Riemannian metric on M. Whether M admits such a metric or not involves its topology and the index of the metric. Whether (M,F) admits a transverse metric involves moreover the dynamics of the foliation and is hereby generally a difficult question. On compact 3-manifolds, the flows admitting a transverse Riemannan metric have been classified by Yves Carrière in the eighties. We wish to classify flows with a transverse Lorentzian metric on the same manifolds -- the signature of the metric is then (1,1). The situation is deeply different, e.g. all algebraic Anosov flows are transversely Lorentzian. We give the classification with an assumption of "transverse completeness" and build a new family of flows showing that this completeness is not always satisfied, unlike in the Riemannian case.
16:50 - 17:40	Thomas Neukirchner (Humboldt Universität Berlin, Berlin, Germany) Examples of compact homogeneous Lorentz manifolds with special holonomy

Participants

Dmitri Alekseevsky

University of Hull, United Kingdom

Said Benayadi

Université de Metz, France

Lionel Berard-Bergery

Université Henri Poincaré Nancy 1, France

Charles Boubel

École normale supérieure de Lyon, France

Shirley Bromberg

Universidad Autonoma Metropolitana, Iztapalapa, Mexico

Michel Cahen

Université Libre de Bruxelles, Belgium

Sergio Console

Università di Torino, Italy

Vicente Cortés Suárez

Universität Hamburg, Germany

Antonio José Di Scala

Politecnico di Torino, Italy

Jost Eschenburg

Universität Augsburg, Germany

José Miguel Figueroa-O'Farrill

University of Edinburgh, United Kingdom

Anton Galaev

Humboldt Universität Berlin, Germany

Sergio Garbiero

Università di Torino, Italy

Michel Goze

Université de Haute Alsace, France

Ernst Heintze

Universität Augsburg, Germany

Sebastian Klein

Universität Köln, Germany

Frank Klinker

Universität Dortmund, Germany

Tom Krantz

Institut Elie Cartan Nancy, France

Margarita Kraus

Johannes Gutenberg-Universität Mainz, Germany

Mario Listing

Universität Dortmund, Germany

Alberto Medina

Université Montpellier 2, France

Thomas Neukirchner

Humboldt Universität Berlin, Germany

Martin Olbrich

Georg-August Universität Göttingen, Germany

Carlos Olmos

Universidad Nacional de Córdoba, Argentina

Peter Quast

Universität Augsburg, Germany

Frank Reidegeld

Universität Dortmund, Germany

Elisabeth Remm

Université de Haute Alsace, France

Lorenz Schwachhöfer

Universität Dortmund, Germany

Hassène Siby

Université Montpellier 2, France

Jürgen Tolksdorf

Max Planck Institute for Mathematics in the Sciences, Germany

Abdelghani Zeghib

École Normale Supérieure de Lyon, France

Scientific Organizers

Helga Baum

Humboldt Universität zu Berlin

Ines Kath

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Administrative Contact

Antje Vandenberg

Max-Planck-Institut für Mathematik in den Naturwissenschaften Contact via Mail