Real hypersurfaces in complex manifolds

Chern, Shiing-Shen; Moser, Jürgen

doi:10.1007/bf02392146

Cited by 778 publications

(474 citation statements)

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“…The generic corresponding situation is the equivalence problem between strictly pseudoconvex hypersurfaces in complex manifolds. Such classifications, initiated and completely treated byÉ.Cartan [13] in complex dimension two, was intensively studied and two important approaches in higher dimension are the theory of normal forms due to SS.Chern-J.Moser [16] and the Fefferman theorem [31] connecting complex and Cauchy-Riemann geometries.…”

Section: Introductionmentioning

confidence: 99%

Some aspects of analysis on almost complex manifolds with boundary

Coupet¹,

Gaussier²,

Sukhov³

2008

J Math Sci

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Section: Introductionmentioning

confidence: 99%

Some aspects of analysis on almost complex manifolds with boundary

Coupet¹,

Gaussier²,

Sukhov³

2008

J Math Sci

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“…Here, S(q) is the Chern curvature tensor [30] of M evaluated at q and c(n) > 0 S is identically zero precisely when M is locally CR equivalent to the sphere. …”

Section: The Cr Yamabe Problemmentioning

confidence: 99%

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Gamara

2017

Arab. J. Math.

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Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings.Mathematics Subject Classification 58E05 · 57R70 · 53C21 · 53C17

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“…It turns out that it exists a canonical H p,q -bundle P over any pseudoconformal manifold M 2n+1 (whose Hermitian structure on Q has signature (p, q)), and a canonical Cartan connection ω : T P → psu(p + 1, q + 1) (where the latter is the Lie algebra of the above mentioned group) [17], [4]. Its curvature measures the obstruction to the construction of a local diffeomorphism Ξ : P → P SU (p + 1, q + 1), for which ω would be the differential (in particular, Ξ would induce a group structure on the universal covering of P , locally isomorphic to P SU (p + 1, q + 1)), and it locally determines the pseudo-conformal structure, see Tanaka [17]; see also [18], [4]; see [9] for a general theory of Cartan connections, and [16] for a general theory of simple graded Lie algebras and G-structures.…”

Section: Sasakian Geometrymentioning

confidence: 99%

“…Its curvature measures the obstruction to the construction of a local diffeomorphism Ξ : P → P SU (p + 1, q + 1), for which ω would be the differential (in particular, Ξ would induce a group structure on the universal covering of P , locally isomorphic to P SU (p + 1, q + 1)), and it locally determines the pseudo-conformal structure, see Tanaka [17]; see also [18], [4]; see [9] for a general theory of Cartan connections, and [16] for a general theory of simple graded Lie algebras and G-structures.…”

Section: Sasakian Geometrymentioning

confidence: 99%

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Normal CR structures on compact 3-manifolds

Belgun¹

2001

Mathematische Zeitschrift

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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...

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Real hypersurfaces in complex manifolds

Cited by 778 publications

References 6 publications

Some aspects of analysis on almost complex manifolds with boundary

Some aspects of analysis on almost complex manifolds with boundary

Methods of the theory of critical points at infinity on Cauchy Riemann manifolds

Normal CR structures on compact 3-manifolds

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