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.
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.
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.
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.
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.
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.
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.
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.)
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.
