Normal C R structures on S3

Belgun, Florin

doi:10.1007/s00209-002-0482-y

Cited by 12 publications

(12 citation statements)

References 17 publications

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“…For a 3-dimensional CR-manifold M , not isomorphic to a sphere, the Sasakian metric is unique, hence the corresponding cone is also unique. Theorem 1.2 and Theorem 1.3 in dimension 3 follow immediately from [Be1], [Be2]. In this paper, we shall always assume that dim M 5.…”

Section: Sasakian Manifolds and Algebraic Conesmentioning

confidence: 98%

“…In dimension 3, F. A. Belgun has shown that all Sasakian manifolds are obtained this way (see [G], [Be1], [Be2]): Theorem 1.10. A strictly pseudoconvex, compact CR-manifold M of dimension 3 admits a Sasakian metric if and only if M is isomorphic to a U (1)-fibration associated with a positive line bundle on a projective orbifold (Example 1.9).…”

Section: Sasakian Geometry and Contact Geometrymentioning

confidence: 99%

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Sasakian structures on CR-manifolds

Verbitsky

2007

Geom Dedicata

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A contact manifold M can be defined as a quotient of a symplectic manifold X by a proper, free action of R >0 , with the symplectic form homogeneous of degree 2. If X is, in addition, Kähler, and its metric is also homogeneous of degree 2, M is called Sasakian. A Sasakian manifold is realized naturally as a level set of a Kähler potential on a complex manifold, hence it is equipped with a pseudoconvex CR-structure. We show that any Sasakian manifold M is CR-diffeomorphic to an S 1 -bundle of unit vectors in a positive line bundle on a projective Kähler orbifold. This induces an embedding from M to an algebraic cone C. We show that this embedding is uniquely defined by the CRstructure. Additionally, we classify the Sasakian metrics on an odd-dimensional sphere equipped with a standard CR-structure.

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Section: Sasakian Manifolds and Algebraic Conesmentioning

confidence: 98%

Section: Sasakian Geometry and Contact Geometrymentioning

confidence: 99%

Sasakian structures on CR-manifolds

Verbitsky

2007

Geom Dedicata

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“…Let N 3 be a complete, simply-connected Sasakian 3-manifold. When N is compact, or more generally when N is co-compact in the sense that there is a compact subset K ⊆ N and a group Γ of isometries of N preserving the Sasakian structure such that the union of h(K) for all h ∈ Γ covers N , then N is classified by Belgun [4], [5], [6]. In particular, it was shown in [6,Theorem 4.5] that after the so-called parallel modification, N 3 can be deformed to one of three standard Lie groups with left invariant Sasakian structures: SU (2), SL(2, C), and N il 3 .…”

Section: It Follows Thatmentioning

confidence: 99%

On Strominger Kähler-like manifolds with degenerate torsion

Yau¹,

Zhao²,

Zheng³

2019

Preprint

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In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold (M n , g) is always pluriclosed, and when the manifold is compact and g is not Kähler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when n = 2, the SKL condition is equivalent to the Vaisman condition. In this paper, we give a classification for compact non-Kähler SKL manifolds in dimension 3 and those with degenerate torsion in higher dimensions. We also present some properties about SKL manifolds in general dimensions, for instance, for any compact non-Kähler SKL manifold, its Kähler form represents a non-trivial Aeppli cohomology class, the metric can never be locally conformal Kähler when n ≥ 3, and the manifold does not admit any Hermitian symplectic metric.

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“…endowed with a constant curvature metric; see e.g. [5] or [20 by isospectrality of P * P and P P * except on kernels. The only remaining term here has to be a constant, while we have a t −1/2 in (5.41).…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

Analytic torsions on contact manifolds

Rumin¹,

Seshadri²

2012

Ann. inst. Fourier

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We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae.Comment: 40 page

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Normal C R structures on S3

Cited by 12 publications

References 17 publications

Sasakian structures on CR-manifolds

Sasakian structures on CR-manifolds

On Strominger Kähler-like manifolds with degenerate torsion

Analytic torsions on contact manifolds

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