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We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.
We obtain a theory of stratified Sternberg spaces thereby extending the theory of cotangent bundle reduction for free actions to the singular case where the action on the base manifold consists of only one orbit type. We find that the symplectic reduced spaces are stratified topological fiber bundles over the cotangent bundle of the orbit space. We also obtain a Poisson stratification of the Sternberg space. To construct the singular Poisson Sternberg space we develop an appropriate theory of singular connections for proper group actions on a single orbit type manifold including a theory of holonomy extending the usual Ambrose– Singer theorem for principal bundles.
The first part of this paper contains a thorough exposition of the proof of the classification of topologically trivial Legendrian knots (i.e., Legendrian knots bounding embedded 2-disks) in tight contact 3-manifolds. These techniques were never published in detail when the classification result was announced over ten years ago. The final part of the present paper contains a systematic discussion of Legendrian knots in overtwisted contact manifolds, and in particular, gives the coarse classification (i.e., classification up to a global contactomorphism) of topologically trivial exceptional Legendrian knots in overtwisted contact $S^3$ according to the values of the invariants tb, r. We show, moreover, that such knots only occur for one of the infinitely many overtwisted contact structures on $S^3$. We remark that our tight classification result also implies that any topologically trivial loose Legendrian knots with same value of (tb, r) in an overtwisted contact 3-manifold are in fact Legendrian isotopic if $tb < 0$.