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 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.

scholarly journals Diastatic entropy and rigidity of complex hyperbolic manifolds

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Roberto Mossa
Keyword(s):  
Lower Bound ◽  
Hyperbolic Manifold ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Metric ◽  
Continuous Map ◽  
The Hyperbolic Metric ◽  
Geometric Rigidity ◽  
Volume Entropy ◽  
Complex Hyperbolic Manifold ◽  
Real Analytic

AbstractLet f : Y → X be a continuous map between a compact real analytic Kähler manifold (Y, g) and a compact complex hyperbolic manifold (X, g0). In this paper we give a lower bound of the diastatic entropy of (Y, g) in terms of the diastatic entropy of (X, g0) and the degree of f . When the lower bound is attained we get geometric rigidity theorems for the diastatic entropy analogous to the ones obtained by G. Besson, G. Courtois and S. Gallot [2] for the volume entropy. As a corollary,when X = Y,we get that the minimal diastatic entropy is achieved if and only if g is isometric to the hyperbolic metric g0.

scholarly journals Convergence of the Chern–Moser–Beloshapka normal forms

2020 ◽  
Vol 2020 (765) ◽  
pp. 205-247
Author(s):  
Bernhard Lamel ◽  
Laurent Stolovitch
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Direct Proof ◽  
Sufficient Condition ◽  
Normalizing Transformation ◽  
General Sufficient Condition ◽  
Real Analytic ◽  
Moser Normal Form

AbstractIn this article, we give a normal form for real-analytic, Levi-nondegenerate submanifolds of{\mathbb{C}^{N}}of codimension{d\geq 1}under the action of formal biholomorphisms. We find a very general sufficient condition on the formal normal form that ensures that the normalizing transformation to this normal form is holomorphic. In the case{d=1}our methods in particular allow us to obtain a new and direct proof of the convergence of the Chern–Moser normal form.

scholarly journals Normal forms for perturbations of systems possessing a Diophantine invariant torus

2017 ◽  
Vol 39 (8) ◽  
pp. 2176-2222 ◽  
Author(s):  
JESSICA ELISA MASSETTI
Keyword(s):  
Vector Fields ◽  
Normal Forms ◽  
Invariant Tori ◽  
Inverse Function ◽  
Hamiltonian Mechanics ◽  
Inverse Function Theorem ◽  
Unified Framework ◽  
Normal Form Theorem ◽  
Real Analytic ◽  
Twist Theorem

We give a new proof of Moser’s 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in analytic class, is flexible and can be adapted to several contexts. This allows us to prove in a unified framework the persistence, up to finitely many parameters, of Diophantine quasi-periodic normally hyperbolic reducible invariant tori for vector fields originating from dissipative generalizations of Hamiltonian mechanics. As a byproduct, generalizations of Herman’s twist theorem and Rüssmann’s translated curve theorem are proved.

scholarly journals Normal forms for nonintegrable almost CR structures

2012 ◽  
Vol 134 (4) ◽  
pp. 915-947 ◽  
Author(s):  
Dmitri Zaitsev
Keyword(s):  
Normal Forms ◽  
Cr Structures

scholarly journals Non-degenerate real hypersurfaces in complex manifolds admitting large groups of pseudo-conformal transformations. I

1976 ◽  
Vol 62 ◽  
pp. 55-96 ◽  
Author(s):  
Keizo Yamaguchi
Keyword(s):  
Complex Manifold ◽  
Real Hypersurface ◽  
Levi Form ◽  
Real Hypersurfaces ◽  
Conformal Transformations ◽  
Complex Manifolds ◽  
Real Analytic ◽  
Large Groups

Let S (resp. S′) be a (real) hypersurface (i.e. a real analytic sub-manifold of codimension 1) of an n-dimensional complex manifold M (resp. M′). A homeomorphism f of S onto S′ is called a pseudo-conformal homeomorphism if it can be extended to a holomorphic homeomorphism of a neighborhood of S in M onto a neighborhood of S′ in M. In case such an f exists, we say that S and S′ are pseudo-conformally equivalent. A hypersurface S is called non-degenerate (index r) if its Levi-form is non-degenerate (and its index is equal to r) at each point of S.

scholarly journals Normal forms for elements of o(p, q) and Hamiltonians with integrals linear in momenta

1992 ◽  
Vol 33 (4) ◽  
pp. 486-507 ◽  
Author(s):  
Gerard Thompson
Keyword(s):  
Dynamical Systems ◽  
Normal Form ◽  
Lie Algebra ◽  
Normal Forms ◽  
First Integral ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Hamiltonian Dynamical Systems ◽  
Matrix Normal

AbstractWe solve the problem of finding a simultaneous matrix normal form for an element of the Lie algebra o(p, q) and the underlying indefinite inner product. The results are used to determine several classes of classical Hamiltonian dynamical systems which possess a first integral linear in the momentum variables.

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