scholarly journals On Propagation of Sphericity of Real Analytic Hypersurfaces across Levi Degenerate Loci

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Joël Merker
Keyword(s):  
Open Subset ◽  
Levi Form ◽  
Dense Open Subset ◽  
Real Analytic Hypersurface ◽  
Analytic Hypersurface ◽  
Real Analytic ◽  
Real Analytic Hypersurfaces

A connected real analytic hypersurface M⊂Cn+1 whose Levi form is nondegenerate in at least one point—hence at every point of some Zariski-dense open subset—is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k,n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having a different signature (l,n-l). Up to signature, pseudosphericity then jumps across the Levi degenerate locus and in particular across the nonminimal locus, if there exists any.

scholarly journals Convergent normal form for real hypersurfaces at a generic Levi-degeneracy

2019 ◽  
Vol 2019 (749) ◽  
pp. 201-225
Author(s):  
Ilya Kossovskiy ◽  
Dmitri Zaitsev
Keyword(s):  
Normal Form ◽  
Moduli Space ◽  
Real Hypersurface ◽  
Convergence Result ◽  
Explicit Description ◽  
Real Hypersurfaces ◽  
Real Analytic ◽  
Real Analytic Hypersurfaces

Abstract We construct a complete convergent normal form for a real hypersurface in {\mathbb{C}^{N}} , {N\geq 2} , at a generic Levi-degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. As an application of the convergence result, we obtain an explicit description of the moduli space of germs of real-analytic hypersurfaces with a generic Levi-degeneracy. As another application, we obtain, in the spirit of the work of Chern and Moser [6], distinguished curves inside the Levi-degeneracy set that we call degenerate chains.

Local Topological Properties of Maps and Open Extensions of Maps

1977 ◽  
Vol 29 (6) ◽  
pp. 1121-1128
Author(s):  
J. K. Kohli
Keyword(s):  
Topological Space ◽  
Open Subset ◽  
Countable Union ◽  
Dense Open Subset ◽  
Topological Properties ◽  
Hausdorff Spaces ◽  
Local Homeomorphism ◽  
Open Map

A σ-discrete set in a topological space is a set which is a countable union of discrete closed subsets. A mapping ƒ : X ⟶ Y from a topological space X into a topological space Y is said to be σ-discrete (countable) if each fibre ƒ-1(y), y ϵ Y is σ-discrete (countable). In 1936, Alexandroff showed that every open map of a bounded multiplicity between Hausdorff spaces is a local homeomorphism on a dense open subset of the domain [2].

On Prime Immersions of S1 into R2

1976 ◽  
Vol 28 (3) ◽  
pp. 589-593
Author(s):  
John R. Martin
Keyword(s):  
Finite Number ◽  
Open Subset ◽  
Double Point ◽  
Dense Open Subset ◽  
Double Points ◽  
Combinatorial Invariant ◽  
Linearly Independent ◽  
Oriented Plane ◽  
Intersection Sequence ◽  
Element Set

A C1-mapping ƒ from the oriented circle S1 into the oriented plane R2 such that f f’ (t) ≠ 0 for all t is called a regular immersion. We call a point p in Im f a double point if f-1(p) is a two element set with the corresponding tangent vectors being linearly independent. A regular immersion which is one-to-one except at a finite number of points whose images are double points is called a normal immersion. The work of Whitney [7], Titus [3] and Verhey [6] shows that the normal immersions form a dense open subset in the space of regular immersions with the usual C1-topology, and can be characterized up to diffeomorphic equivalence by a combinatorial invariant called the intersection sequence.

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