Special Symplectic Connections

By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-K\"ahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.Comment: 35 pages, no figures. Exposition improved, some minor errors corrected. Version to be published by Jour.Diff.Geo

Nonnegative curvature on disk bundles

The search for manifolds of nonnegative curvature1 is one of the classical problems in Riemannian geometry. While general obstructions are scarce, there are relatively few general classes of examples and construction methods. Hence, it is unclear how large one should expect the class of closed manifolds admitting a nonnegatively curved metric to be. For a survey of known examples, see e.g. [Z]. Apart from taking products, there are only two general methods to construct new nonnegatively curved metrics out of given spaces. One is the use of Riemannian submersions which non-decrease curvature by O’Neill’s formula. The other is the glueing of two manifolds (which we call halves) along their common boundary. Typically, the boundary of each half is assumed to be totally geodesic or, slightly more restrictive, a collar metric. This in turn implies by the Soul theorem ([CG]) that each half is the total space of a disk bundle over a totally geodesic closed submanifold. In addition, the glueing map of the two boundaries must be an isometry. While many examples can be constructed by such a glueing, its application is still limited. On the one hand, there is not too much known on the question which disk bundles over a nonnegatively curved compact manifold admit collar metrics of nonnegative curvature, and on the other hand, even if such metrics exist, the metric on the boundary is not arbitrary. Thus, glueing together two such disk bundles to a nonnegatively curved closed manifold is possible in special situations only. For instance, if the disk bundle is homogeneous, then there always exist invariant nonnegatively curved collar metrics. However, the metric on the boundary of such a collar metric is restricted due to the existence of certain parallel Killing fields by a result of Perelman ([P]). In this article, we will give a survey of known examples and describe some recent results which illustrate the difficulty in finding metrics on disk bundles which are suitable for this glueing construction

On the Incompleteness of Berger's List of Holonomy Representations

In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Berger's classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with these new holonomies. We also deduce some striking facts about their global behaviour.Comment: 20 pages, AMS-LaTeX, no figure

On the Solvability of the Transvection group of Extrinsic Symplectic Symmetric Spaces

Let $M$ be a symplectic symmetric space, and let $\imath : M \to V$ be an extrinsic symplectic symmetric immersion, i.e., $(V, \Omega)$ is a symplectic vector space and $\imath$ is an injective symplectic immersion such that for each point $p \in M$, the geodesic symmetry in $p$ is compatible with the reflection in the affine normal space at $\imath(p)$. We show that the existence of such an immersion implies that the transvection group of $M$ is solvable.Comment: 15 page

Special connections on symplectic manifolds

On a given symplectic manifold, there are many symplectic connections, i.e. torsion free connections w.r.t. which the symplectic form is parallel. We call such a connection special if it is either the Levi-Civita connection of a Bochner-Kähler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. We link these special connections to parabolic contact geometry, showing that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of rigidity results and other global prop-erties