Riemannian metrics with lower curvature bounds. Special symplectic connections and symplectic realizations.

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Riemannian metrics with lower curvature bounds: Among the fundamental questions in differential geometry is the determination of obstructions to the existence or the description of new examples of manifolds with given lower curvature bounds. In particular, it is of interest to investigate obstructions to or examples of metrics with positive, nonnegative or almost nonnegative sectional or Ricci curvature. The class of manifolds which are of interest here are those which admit a group action of low cohomogeneity, in particular those of cohomogeneity at most two.Symplectic connections and symplectic realizations: A connection an a manifold is the description of the parallel transport of tangent vectors along differentiable paths. If the manifold is symplectic, then we call the connection symplectic if parallel transport preserves the symplectic form. We investigate symplectic connections whose curvature satisfies certain conditions, namely either the vanishing of some part of the curvature, or restrictions on its holonomy group. There is a canonical method to construct such connections locally. This method is based on the local existence of symplectic realization of certain Poisson structures related to quaternionic symmetric spaces. The aim is to determine in which case there can be a global realization and hence when the local obstruction methods can be extended globally. In particular, it is important to decide if the Poisson structures involved admit a realization by a symplectic groupoid.