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Abstract. We study normal CR compact manifolds in dimension 3. For a choice of a CR Reeb vector field, we associate a Sasakian metric on them, and we classify those metrics. As a consequence, the underlying manifolds are topologically finite quotiens of S 3 or of a circle bundle over a Riemann surface of positive genus. In the latter case, we prove that their CR automorphisms group is a finite extension of S 1 , and we classify the normal CR structures on these manifolds. IntroductionAnalogs of complex manifolds in odd dimensions, pseudo-conformal CR manifolds are particular contact manifolds, with a complex structure on the corresponding distribution of hyperplanes, satisfying an integrability condition (see Section 2). Contrary to complex geometry, CR geometry is locally determined by a finite system of local invariants (like in the cases of conformal or projective structures), [17], [9], [16]. Therefore the space of locally non-isomorphic CR structures is a space with infinitely many parameters.In this paper, we focus our attention on normal CR manifolds, which admit global Reeb vector fields preserving the CR structure, in particular their CR automorphisms group has dimension at least 1. Our main result is that, for a compact normal CR 3-manifold, which is topologically not a quotient of S 3 , this CR automorphisms group is a finite extension of a circle, thus the Reeb vector field is unique up to a constant (Section 4, Theorem 2). This, together with the classification of Sasakian compact 3-manifolds (see Section 3), allows us to obtain the classification of normal CR structures on these manifolds (Section 4, Corollary 4).The question of classifying compact CR manifolds has first been solved in situations with a high order of local symmetry: the classification of flat compact CR manifolds, where the local CR automorphism group is P SU (n + 1, 1) (if the manifold has dimension 2n + 1), is due to E. Cartan [3] and to D. Burns and S. Shnider [2]; in dimension 3, homogeneous, simplyconnected, CR manifolds are either flat or (3-dimensional) Lie groups, and have been classified by E. Cartan [3] (see also [5]). In this case, there is no intermediate symmetry because E. Cartan has showed that a homogeneous CR manifold whose CR automorphism group has dimension greater than 3 is automatically flat.In dimension 3, the normal CR structures are always deformations of a flat one (Theorem 1, see also [1]), and the key point is that, for a CR Reeb vector field T , they admit compatible Sasakian metrics, for which T is 1 2 FLORIN ALEXANDRU BELGUN Killing (see Section 2 for details); these metrics are closely related to locally conformally Kähler metrics with parallel Lee form, natural analogs of Kähler structures on non-symplectic complex manifolds [19].Topologically, every compact normal CR (or Sasakian) 3-manifold is a Seifert fibration (Proposition 5, see also [8], [7] and [1]), but it turns out that the Sasakian structures themselves can be explicitly described on these manifolds: Theorem 1, Proposition 5 (these are ext...

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form (p ≥ 2) if and only if it isometric to a Riemannian product S k × N , where S k is a round sphere and k > p.2000 Mathematics Subject Classification: Primary 53C55, 58J50.

Motivated by the study of Weyl structures on conformal manifolds admitting parallel weightless forms, we define the notion of conformal product of conformal structures and study its basic properties. We obtain a classification of Weyl manifolds carrying parallel forms, and we use it to investigate the holonomy of the adapted Weyl connection on conformal products. As an application we describe a new class of Einstein-Weyl manifolds of dimension 4.2000 Mathematics Subject Classification: Primary 53A30, 53C05, 53C29.

We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemin-Sternberg. We also prove similar results for the symplectic moment map (defined on the minimal presentation) whose image is then a convex cone. In the special case of a compact toric Vaisman manifold, we obtain a structure theorem. Preliminaries on lcs, lcK and Vaisman manifoldsLet (M, ω) be an almost symplectic manifold of real dimension greater than 2, where ω is a non-degenerate 2-form. Often ω := g(J·, ·) will be the fundamental form of an (almost) Hermitian metric g on (M, J), where J : T M → T M is an (almost) complex structure on M . We will usually consider the complex case, when J is integrable.If every point of M admits a neighborhood U and a smooth function f U : U → R such that the two-form e −fU ω| U is closed, we call (M, ω) a locally conformally symplectic manifold (lcs). If ω is the fundamental (or Kähler) form of a Hermitian manifold (M, g, J), then we call it a locally conformally Kähler manifold (lcK).From the definition, it follows that the local 1-forms df U glue together to a global 1-form θ, called the Lee form, satisfying on M