No abstract

Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with H 1 (M, Z) = 0 which is K-contact but which carries no semi-regular Sasakian structures.2010 Mathematics Subject Classification. 53C25, 53D35, 57R17, 14J25.

We construct a compact simply connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply connected 7-dimensional Sasakian manifold has vanishing cup product H 2 × H 2 → H 4 and that it is formal if and only if all its triple Massey products vanish.

We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this, we produce a method of constructing simply connected K‐contact non‐Sasakian manifolds. On the other hand, for every n⩾3, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension 2n+1 that are non‐formal. They are non‐formal because they have a non‐zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold.

Topology and its Applications 127 (2003) 33-45. doi:10.1016/S0166-8641(02)00041-XReceived by publisher: 2000-08-30Harvest Date: 2016-01-04 12:21:28DOI: 10.1016/S0166-8641(02)00041-XPage Range: 33-4

We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Sternberg, Weinstein and Lerman

The aim of the present paper is to investigate new classes of symplectically fat fibre bundles. We prove a general existence theorem for fat vectors with respect to the canonical invariant connections. Based on this result we give new proofs of some constructions of symplectic structures. This includes twistor bundles and locally homogeneous complex manifolds. The proofs are conceptually simpler and allow for obtaining more general results.Comment: 23 pages; no figure

The purpose of this article is to introduce and investigate properties of a tool (the a-hyperbolic rank) which enables us to obtain new examples of homogeneous spaces G/H which admit and do not admit a discontinuous action of a non virtually-abelian discrete subgroup. We achieve this goal by exploring in greater detail the technique of adjoint orbits developed by Okuda combined with the well-known conditions of Benoist. We find easy-to-check conditions on G and H expressed directly in terms of the Satake diagrams of the corresponding Lie algebras, in cases when G is a real form of a complex Lie group of type A n , D 2k+1 or E 6 . One of the advantages of this approach is the fact, that we don't need to know the embedding of H into G. Using the a-hyperbolic rank we also show, that the homogeneous space E 6 IV /H of reductive type admits a discontinuous action of a non virtually abelian discrete subgroup if and only if H is compact. Also, inspired by the work of Okuda on symmetric spaces G/H we find a list of simply connected 3-symmetric spaces admitting a discontinuous action of a non virtually-abelian discrete subgroup. This list yields almost complete classification of such spaces (there is one exception).