scholarly journals WHAT IS...an Affine Sphere?

2012 ◽  
Vol 59 (03) ◽  
pp. 1
Author(s):  
Daniel J. F. Fox
Keyword(s):  
Affine Sphere

scholarly journals Characterization of affine ruled surfaces

1997 ◽  
Vol 39 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Włodzimierz Jelonek
Keyword(s):  
Shape Operator ◽  
Ruled Surfaces ◽  
Affine Sphere ◽  
Affine Surface ◽  
Codazzi Tensor ◽  
Difference Tensor ◽  
The Difference ◽  
In The Beginning ◽  
Affine Metric

The aim of this paper is to give certain conditions characterizing ruled affine surfaces in terms of the Blaschke structure (∇, h, S) induced on a surface (M, f) in ℝ3. The investigation of affine ruled surfaces was started by W. Blaschke in the beginning of our century (see [1]). The description of affine ruled surfaces can be also found in the book [11], [3] and [7]. Ruled extremal surfaces are described in [9]. We show in the present paper that a shape operator S is a Codazzi tensor with respect to the Levi-Civita connection ∇ of affine metric h if and only if (M, f) is an affine sphere or a ruled surface. Affine surfaces with ∇S = 0 are described in [2] (see also [4]). We also show that a surface which is not an affine sphere is ruled iff im(S - HI) =ker(S - HI) and ket(S - HI) ⊂ ker dH. Finally we prove that an affine surface with indefinite affine metric is a ruled affine sphere if and only if the difference tensor K is a Codazzi tensor with respect to ∇.

scholarly journals Affine Sphere Relativity

2016 ◽  
Vol 350 (2) ◽  
pp. 749-801 ◽  
Author(s):  
E. Minguzzi
Keyword(s):  
Affine Sphere

scholarly journals Affine spheres with prescribed Blaschke metric

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 5967-5975
Author(s):  
Barbara Opozda
Keyword(s):  
Gaussian Curvature ◽  
Harmonic Functions ◽  
Necessary Condition ◽  
Dimensional Manifold ◽  
Affine Sphere ◽  
Metric Tensor ◽  
Affine Spheres ◽  
Pick Invariant ◽  
Sufficient And Necessary Condition

It is proved that the equality ?ln |k-?| = 6k, where k is the Gaussian curvature of a metric tensor 1 on a 2-dimensional manifold is a sufficient and necessary condition for local realizability of the metric as the Blaschke metric of some affine sphere. Consequently, the set of all improper local affine spheres with nowhere-vanishing Pick invariant can be parametrized by harmonic functions.

scholarly journals Affine focal sets of codimension-2 submanifolds contained in hypersurfaces

2018 ◽  
Vol 148 (5) ◽  
pp. 995-1016
Author(s):  
Marcos Craizer ◽  
Marcelo J. Saia ◽  
Luis F. Sánchez
Keyword(s):  
Vector Field ◽  
Single Line ◽  
Position Vector ◽  
General Description ◽  
Affine Sphere ◽  
Affine Normal ◽  
Darboux Vector ◽  
Focal Set ◽  
Regular Hypersurface ◽  
Affine Metric

In this paper we study the affine focal set, which is the bifurcation set of the affine distance to submanifolds Nn contained in hypersurfaces Mn+1 of the (n + 2)-space. We give conditions under which this affine focal set is a regular hypersurface and, for curves in 3-space, we describe its stable singularities. For a given Darboux vector field ξ of the immersion N ⊂ M, one can define the affine metric g and the affine normal plane bundle . We prove that the g-Laplacian of the position vector belongs to if and only if ξ is parallel. For umbilic and normally flat immersions, the affine focal set reduces to a single line. Submanifolds contained in hyperplanes or hyperquadrics are always normally flat. For N contained in a hyperplane L, we show that N ⊂ M is umbilic if and only if N ⊂ L is an affine sphere and the envelope of tangent spaces is a cone. For M hyperquadric, we prove that N ⊂ M is umbilic if and only if N is contained in a hyperplane. The main result of the paper is a general description of the umbilic and normally flat immersions: given a hypersurface f and a point O in the (n + 1)-space, the immersion (ν, ν · (f − O)), where ν is the co-normal of f, is umbilic and normally flat, and conversely, any umbilic and normally flat immersion is of this type.

scholarly journals Pseudo-Isotropic Centro-Affine Lorentzian Surfaces

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1284
Author(s):  
Olivier Birembaux
Keyword(s):  
Complete Classification ◽  
Affine Sphere ◽  
Difference Tensor

In this paper, we study centro-affine Lorentzian surfaces M2 in ℝ3 which have pseudo-isotropic or lightlike pseudo-isotropic difference tensor. We first show that M2 is pseudo-isotropic if and only if the Tchebychev form T=0. In that case, M2 is a an equi-affine sphere. Next, we will get a complete classification of centro-affine Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isotropic.

Sign in / Sign up

Export Citation Format

Share Document