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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

In the complex-Riemannian framework we show that a conformal manifold containing a compact, simply-connected, null-geodesic is conformally¯at. In dimension 3 we use the LeBrun correspondence, that views a conformal 3-manifold as the conformal in®nity of a selfdual four-manifold. We also ®nd a relation between the conformal invariants of the conformal in®nity and its ambient.Our main result (section 4, Theorem 4) states that, if a conformal complex n-manifold admits a rational curve as a null-geodesic, then it is conformally¯at (see also [17] for Partially supported by the SFB 288 of the DFG.Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/1/15 10:56 AM the case of a complex projective manifold). The proof uses the properties of Jacobi ®elds along the considered compact, simply-connected, null-geodesic: namely, we compute the normal bundle of a compact, simply-connected, null-geodesic, and we show that the small deformations of the latter as a compact curve, or as a null-geodesic, coincide (section 4, Proposition 5). In addition to that, we use, for the (more di½cult) case of dimension 3, a criterion for conformal¯atness from [2], and we apply it to a locally de®ned, by the LeBrun correspondence (see below), self-dual ambient.The other topic of this paper uses implicitly another application of twistor theory: It has been shown by LeBrun [8], [9], that any conformal 3-manifold can be locally realized as the conformal in®nity of a self-dual Einstein (with non-zero scalar curvature) 4-manifold. We have, thus, a local correspondence assigning to a conformal structure in dimension 3 a self-dual Einstein metric in dimension 4, which we call the LeBrun correspondence.As conformal structures of both manifolds are encoded in the complex, resp. CR, structure of their twistor spaces, they are implicitly related, for example if the 3-manifold M is conformally¯at, its ambient space N equally is. It is, however, di½cult to obtain an explicit relation between the conformal invariants of M and those of N by twistorial methods, as there is no simple expression of the Cotton-York tensor of M 3 in twistorial terms, and the twistorial interpretation of the Weyl tensor of N 4 is highly non-linear [2].In this paper we ®nd a relation between these two conformal invariants of the manifolds involved in the LeBrun correspondence, or, more generally, of an umbilic submanifold M 3 and of its self-dual ambient space N 4 . It appears that the Weyl tensor of N 4 identically vanishes along M 3 , and thus the Cotton-York tensor of N 4 , restricted to M 3 , is conformally invariant and can be identi®ed with the Cotton-York tensor of M 3 ; in this case, it is also equal to the normal derivative of the Weyl tensor of N 4 (section 3, Theorem 1). This gives conditions for an open self-dual 4-manifold to admit a conformal in®nity.The paper is organized as follows: in section 2 we recall a few basic facts about complex-Riemannian and -conformal geometry, in section 3 we relate the conformal invaria...

Abstract. A locally metric connection on a smooth manifold M is a torsion-free connection D on T M with compact restricted holonomy group Hol 0 (D). If the holonomy representation of such a connection is irreducible, then D preserves a conformal structure on M . Under some natural geometric assumption on the life-time of incomplete geodesics, we prove that conversely, a locally metric connection D preserving a conformal structure on a compact manifold M has irreducible holonomy representation, unless Hol 0 (D) = 0 or D is the Levi-Civita connection of a Riemannian metric on M . This result generalizes Gallot's theorem on the irreducibility of Riemannian cones to a much wider class of connections. As an application, we give the geometric description of compact conformal manifolds carrying a tame closed Weyl connection with non-generic holonomy.2010 Mathematics Subject Classification: Primary 53A30, 53C05, 53C29.

We classify the normal CR structures on S 3 and their automorphism groups. Together with [2], this closes the classification of normal CR structures on compact 3-manifolds. We give a criterion to compare two normal CR structures, and we show that the underlying contact structure is, up to diffeomorphism, unique. (2000): 53A40, 53C25, 53D10. Mathematics Subject Classification