Inductive schemes for the complete classification of affine hypersurfaces with parallel second fundamental form

2011 ◽  
Vol 52 (1) ◽  
pp. 51-73 ◽  
Author(s):  
Salvador Gigena
Keyword(s):  
Fundamental Form ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Parallel Second Fundamental Form

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 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.

LORENTZIAN SYMMETRIC THREE-SPACES AND THE CLASSIFICATION OF THEIR PARALLEL SURFACES

2009 ◽  
Vol 20 (10) ◽  
pp. 1185-1205 ◽  
Author(s):  
G. CALVARUSO ◽  
J. VAN DER VEKEN
Keyword(s):  
Vector Field ◽  
Fundamental Form ◽  
Homogeneous Spaces ◽  
Three Dimensional ◽  
Second Fundamental Form ◽  
Parallel Second Fundamental Form ◽  
Parallel Surfaces ◽  
Riemannian Case ◽  
Lorentzian Homogeneous Spaces

We describe a global model for Lorentzian symmetric three-spaces admitting a parallel null vector field, and classify completely the surfaces with parallel second fundamental form in all Lorentzian symmetric three-spaces. Interesting differences arise with respect to the Riemannian case studied in [2]. Our results complete the classification of parallel surfaces in all three-dimensional Lorentzian homogeneous spaces.

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 CR-hypersurfaces of the six-dimensional sphere

1994 ◽  
Vol 17 (1) ◽  
pp. 197-200
Author(s):  
M. A. Bashir
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Dimensional Sphere ◽  
Totally Umbilical ◽  
Parallel Second Fundamental Form ◽  
Extrinsic Sphere

We proved that there does not exist a properCR-hypersurface ofS6with parallel second fundamental form. As a result of this we showed thatS6does not admit a properCR-totally umbilical hypersurface. We also proved that an Einstein properCR-hypersurface ofS6is an extrinsic sphere.

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