# Transformation Groups in Pseudo-Riemannian Geometry

## Abstracts

### Einstein symmetric spaces

**Dmitri Alekseevsky** *(University of Hull, United Kingdom)*

Thursday, June 29, 2006

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.

### Metric Lie superalgebras with reductive even parts

**Said Benayadi** *(Université de Metz, France)*

Friday, June 30, 2006

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.

### Lorentzian flows on compact 3-manifolds

**Charles Boubel** *(École normale supérieure de Lyon, France)*

Saturday, July 01, 2006

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.

### Submanifolds and holonomy (I)

**Sergio Console** *(Università di Torino, Italy)*

Thursday, June 29, 2006

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.

### The twistor spaces of a para-quaternionic Kähler manifold

**Vicente Cortés Suárez** *(Universität Hamburg, Germany)*

Saturday, July 01, 2006

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 and a -invariant
distribution . We study the conditions for the integrability of
and for the (para-)holomorphicity of . Then we apply this
theory to para-quaternionic Kähler manifolds of non-zero scalar curvature,
which admit two natural twistor spaces ,
, such that Id. We prove that in both
cases is integrable (recovering results of Blair, Davidov and
Muskarov) and that defines a holomorphic () or
para-holomorphic () contact structure. Furthermore, we determine
all the solutions of the Einstein equation for the canonical one-parameter
family of pseudo-Riemannian metrics on . In particular, we find that
there is a unique Kähler-Einstein () or para-Kähler-Einstein
() metric. Finally, we prove that any Kähler or para-Kähler
submanifold of a para-quaternionic Kähler manifold is minimal and describe all
such submanifolds in terms of complex (), respectively,
para-complex () submanifolds of tangent to the contact
distribution. (This is joint work with Dmitri Alekseevsky.)

### String backgrounds from Lie groups: beyond the WZW model

**José Miguel Figueroa-O'Farrill** *(University of Edinburgh, United Kingdom)*

Friday, June 30, 2006

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.

### Infinite dimensional symmetric spaces of Kac-Moody type

**Ernst Heintze** *(Universität Augsburg, Germany)*

Saturday, July 01, 2006

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.

### Submanifold and holonomy (II)

**Carlos Olmos** *(Universidad Nacional de Córdoba, Argentina)*

Thursday, June 29, 2006

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.

### On the conformal group of Finsler manifolds

**Abdelghani Zeghib ** *(École Normale Supérieure de Lyon, France)*

Saturday, July 01, 2006

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.

## Date and Location

**June 29 - July 01, 2006**

Max Planck Institute for Mathematics in the Sciences

Inselstraße 22

04103 Leipzig

Germany

see travel instructions

## Scientific Organizers

**Helga Baum**

Humboldt Universität zu Berlin

Institut für Mathematik

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**Ines Kath**

Max Planck Institute for Mathematics in the Sciences

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## Administrative Contact

**Antje Vandenberg**

Max Planck Institute for Mathematics in the Sciences

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