Extension of CR structures on pseudoconvex CR-manifolds with comparable eigenvalues of the Levi-form

2021 ◽  
pp. 1-33
Author(s):  
Sanghyun Cho
Keyword(s):  
Levi Form ◽  
Cr Manifolds ◽  
Cr Structures

scholarly journals Para-CR structures on almost paracontact metric manifolds

2014 ◽  
Vol 20 (2) ◽  
Author(s):  
Joanna Wełyczko
Keyword(s):  
Sectional Curvature ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Cr Manifolds ◽  
Cosymplectic Manifolds ◽  
Cr Structures ◽  
Necessary And Sufficient

AbstractAlmost paracontact metric manifolds are the famous examples of almost para-CR manifolds. We find necessary and sufficient conditions for such manifolds to be para-CR. Next we examine these conditions in certain subclasses of almost paracontact metric manifolds. Especially, it is shown that normal almost paracontact metric manifolds are para-CR. We establish necessary and sufficient conditions for paracontact metric manifolds as well as for almost para-cosymplectic manifolds to be para-CR. We find also basic curvature identities for para-CR paracontact metric manifolds and study their consequences. Among others, we prove that any para-CR paracontact metric manifold of constant sectional curvature and of dimension greater than 3 must be para-Sasakian and its curvature equal to -1. The last assertion does not hold in dimension 3. We show that a conformally flat para-Sasakian manifold is of constant sectional curvature equal to -1. New classes of examples of para-CR manifolds are established.

scholarly journals Hölder estimates on CR manifolds with a diagonalizable Levi form

1990 ◽  
Vol 84 (1) ◽  
pp. 1-90 ◽  
Author(s):  
C.L Fefferman ◽  
J.J Kohn ◽  
M Machedon
Keyword(s):  
Levi Form ◽  
Cr Manifolds ◽  
Hölder Estimates

scholarly journals Contact Semi-Riemannian Structures in CR Geometry: Some Aspects

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 6 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifolds ◽  
Integrability Condition ◽  
Cr Manifolds ◽  
Cr Geometry ◽  
Cr Structure ◽  
Cr Structures ◽  
Similarities And Differences ◽  
One To One ◽  
Riemannian Case

There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X - i J X , X ∈ H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case.

scholarly journals INVARIANTS OF ELLIPTIC AND HYPERBOLIC CR-STRUCTURES OF CODIMENSION 2

1999 ◽  
Vol 10 (01) ◽  
pp. 1-52 ◽  
Author(s):  
V. V. EZHOV ◽  
A. V. ISAEV ◽  
G. SCHMALZ
Keyword(s):  
Lie Algebra ◽  
Normal Forms ◽  
Levi Form ◽  
Hyperbolic Manifolds ◽  
Cartan Connection ◽  
Natural Class ◽  
Cr Structures ◽  
Real Analytic ◽  
Flat Manifolds ◽  
Infinitesimal Automorphisms

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves "almost" like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of "semi-flat" manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

scholarly journals ON HOMOGENEOUS CR MANIFOLDS AND THEIR CR ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1199-1214
Author(s):  
ANDREA ALTOMANI ◽  
COSTANTINO MEDORI
Keyword(s):  
Lie Group ◽  
Levi Form ◽  
Flag Manifold ◽  
Semisimple Lie Group ◽  
Real Form ◽  
Cr Manifolds ◽  
Geometrical Properties ◽  
Homogeneous Cr Manifolds

In this paper we show some results on homogeneous CR manifolds, proved by introducing their associated CR algebras. In particular, we give different notions of nondegeneracy (generalizing the usual notion for the Levi form) which correspond to geometrical properties for the corresponding manifolds. We also give distinguished equivariant CR fibrations for homogeneous CR manifolds. In the second part of the paper we apply these results to minimal orbits for the action of a real form of a semisimple Lie group Ĝ on a flag manifold Ĝ/Q.

scholarly journals Extension of CR structures on three dimensional pseudoconvex CR manifolds

1998 ◽  
Vol 152 ◽  
pp. 97-129 ◽  
Author(s):  
Sanghyun Cho
Keyword(s):  
Finite Type ◽  
Complex Structure ◽  
Three Dimensional ◽  
Concave Side ◽  
Almost Complex Structure ◽  
Cr Manifolds ◽  
Almost Complex ◽  
Cr Structures ◽  
The Given

Abstract.Let be a smoothly bounded orientable pseudoconvex CR manifold of finite type and dimℝM = 3. Then we extend the given CR structure on M to an integrable almost complex structure on which is the concave side of M and M ⊂

scholarly journals Blow-ups and infinitesimal automorphisms of CR-manifolds

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1701-1724
Author(s):  
Boris Kruglikov
Keyword(s):  
Lie Group ◽  
Blow Up ◽  
Symmetry Algebra ◽  
Levi Form ◽  
Cr Manifolds ◽  
Parabolic Subalgebras ◽  
Degeneracy Locus ◽  
Real Analytic ◽  
Infinitesimal Automorphisms

Abstract For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

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