scholarly journals Five-Dimensional Contact CR-Submanifolds in S 7 ( 1 )

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1278
Author(s):  
Mirjana Djorić ◽  
Marian Ioan Munteanu
Keyword(s):  
Contact Structure ◽  
Sasakian Manifold ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Remarkable Property ◽  
Dimensional Unit ◽  
Products Of Spheres ◽  
Kählerian Manifolds ◽  
Cr Submanifolds

Due to the remarkable property of the seven-dimensional unit sphere to be a Sasakian manifold with the almost contact structure (φ,ξ,η), we study its five-dimensional contact CR-submanifolds, which are the analogue of CR-submanifolds in (almost) Kählerian manifolds. In the case when the structure vector field ξ is tangent to M, the tangent bundle of contact CR-submanifold M can be decomposed as T(M)=H(M)⊕E(M)⊕Rξ, where H(M) is invariant and E(M) is anti-invariant with respect to φ. On this occasion we obtain a complete classification of five-dimensional proper contact CR-submanifolds in S7(1) whose second fundamental form restricted to H(M) and E(M) vanishes identically and we prove that they can be decomposed as (multiply) warped products of spheres.

scholarly journals Almost Complex Surfaces in the Nearly Kähler SL(2,ℝ) × SL(2,ℝ)

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1160
Author(s):  
Elsa Ghandour ◽  
Luc Vrancken
Keyword(s):  
Complete Classification ◽  
Second Fundamental Form ◽  
Complex Surfaces ◽  
Kähler Structure ◽  
Totally Geodesic ◽  
Parallel Second Fundamental Form ◽  
Almost Complex ◽  
Nearly Kähler ◽  
Nearly Kähler Structure

The space S L ( 2 , R ) × S L ( 2 , R ) admits a natural homogeneous pseudo-Riemannian nearly Kähler structure. We investigate almost complex surfaces in this space. In particular, we obtain a complete classification of the totally geodesic almost complex surfaces and of the almost complex surfaces with parallel second fundamental form.

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Space Form ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Normal Space ◽  
Space Forms ◽  
Parallel Surface ◽  
Parallel Surfaces ◽  
Point To Point ◽  
Lorentz Surface

AbstractA Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $$ \mathbb{E}_2^4 $$ and in neutral pseudo 4-sphere S 24 (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H 24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H 24 (−1). Conversely, every parallel Lorentz surface in H 24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.

scholarly journals Classification of Möbius Isoparametric Hypersurfaces in 4

2005 ◽  
Vol 179 ◽  
pp. 147-162 ◽  
Author(s):  
Zejun Hu ◽  
Haizhong Li
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Isoparametric Hypersurface ◽  
Isoparametric Hypersurfaces ◽  
Möbius Form ◽  
Möbius Geometry ◽  
Dimensional Unit ◽  
Dupin Hypersurfaces ◽  
Möbius Second Fundamental Form

AbstractLet Mn be an immersed umbilic-free hypersurface in the (n + 1)-dimensional unit sphere n+1, then Mn is associated with a so-called Möbius metric g, a Möbius second fundamental form B and a Möbius form Φ which are invariants of Mn under the Möbius transformation group of n+1. A classical theorem of Möbius geometry states that Mn (n ≥ 3) is in fact characterized by g and B up to Möbius equivalence. A Möbius isoparametric hypersurface is defined by satisfying two conditions: (1) Φ ≡ 0; (2) All the eigenvalues of B with respect to g are constants. Note that Euclidean isoparametric hyper-surfaces are automatically Möbius isoparametric, whereas the latter are Dupin hypersurfaces.In this paper, we prove that a Möbius isoparametric hypersurface in 4 is either of parallel Möbius second fundamental form or Möbius equivalent to a tube of constant radius over a standard Veronese embedding of ℝP2 into 4. The classification of hypersurfaces in n+1 (n ≥ 2) with parallel Möbius second fundamental form has been accomplished in our previous paper [6]. The present result is a counterpart of Pinkall’s classification for Dupin hypersurfaces in 4 up to Lie equivalence.

scholarly journals LEGENDRIAN HELIX AND CABLE LINKS

2010 ◽  
Vol 12 (03) ◽  
pp. 487-500 ◽  
Author(s):  
FAN DING ◽  
HANSJÖRG GEIGES
Keyword(s):  
Contact Structure ◽  
Complete Classification ◽  
Jet Space ◽  
Link Type ◽  
Two Component ◽  
Circular Helix

Traynor ([11]) has described an example of a two-component Legendrian "circular helix link" Λ0 ⊔ Λ1 in the 1-jet space J1(S1) of the circle (with its canonical contact structure) that is topologically but not Legendrian isotopic to the link Λ1 ⊔ Λ0. We give a complete classification of the Legendrian realizations of this topological link type, as well as all other "cable links" in J1(S1).

COMPLETE CLASSIFICATION OF PARALLEL LORENTZIAN SURFACES IN LORENTZIAN COMPLEX SPACE FORMS

2010 ◽  
Vol 21 (05) ◽  
pp. 665-686 ◽  
Author(s):  
BANG-YEN CHEN ◽  
FRANKI DILLEN ◽  
JOERI VAN DER VEKEN
Keyword(s):  
Riemannian Manifold ◽  
Fundamental Form ◽  
Complex Space ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Space Forms ◽  
Complex Dimension ◽  
Point To Point ◽  
Complex Space Forms

A surface of a pseudo-Riemannian manifold is called parallel if its second fundamental form is parallel with respect to the Van der Waerden–Bortolotti connection. Such surfaces are fundamental since the extrinsic invariants of the surfaces do no change from point to point. In this article, we completely classify parallel Lorentzian surfaces in Lorentzian complex space forms of complex dimension two.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

scholarly journals Characteristic cohomology and observables in higher spin gravity

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Alexey Sharapov ◽  
Evgeny Skvortsov
Keyword(s):  
Higher Spin Gravity ◽  
Complete Classification ◽  
Gravity Models ◽  
Simons Theory ◽  
Surface Charges ◽  
Higher Spin ◽  
Dynamical Invariants ◽  
Chern Simons ◽  
Conserved Currents

Abstract We give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

scholarly journals Nil-clean quadratic elements

2017 ◽  
Vol 16 (10) ◽  
pp. 1750197 ◽  
Author(s):  
Janez Šter
Keyword(s):  
Complete Classification ◽  
Strong Condition ◽  
Clean Rings

We provide a strong condition holding for nil-clean quadratic elements in any ring. In particular, our result implies that every nil-clean involution in a ring is unipotent. As a consequence, we give a complete classification of weakly nil-clean rings introduced recently in [Breaz, Danchev and Zhou, Rings in which every element is either a sum or a difference of a nilpotent and an idempotent, J. Algebra Appl. 15 (2016) 1650148, doi: 10.1142/S0219498816501486].

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