A cylinder theorem for a (1, 1)-geodesic affine hypersurface with non-flat induced connection

2009 ◽  
Vol 144 (1) ◽  
pp. 61-69 ◽  
Author(s):  
Sanae Kurosu
Keyword(s):  
Affine Hypersurface ◽  
Cylinder Theorem

scholarly journals Almost vanishing polynomials and an application to the Hough transform

2014 ◽  
Vol 13 (08) ◽  
pp. 1450057 ◽  
Author(s):  
Maria-Laura Torrente ◽  
Mauro C. Beltrametti
Keyword(s):  
Hough Transform ◽  
Affine Space ◽  
Bounded Region ◽  
Pattern Recognition Technique ◽  
Local Study ◽  
Affine Hypersurface ◽  
Automated Recognition ◽  
Real Affine ◽  
Necessary And Sufficient ◽  
Recognition Technique

We consider the problem of deciding whether or not an affine hypersurface of equation f = 0, where f = f(x1, …, xn) is a polynomial in ℝ[x1, …, xn], crosses a bounded region 𝒯 of the real affine space 𝔸n. We perform a local study of the problem, and provide both necessary and sufficient numerical conditions to answer the question. Our conditions are based on the evaluation of f at a point p ∈ 𝒯, and derive from the analysis of the differential geometric properties of the hypersurface z = f(x1, …, xn) at p. We discuss an application of our results in the context of the Hough transform, a pattern recognition technique for the automated recognition of curves in images.

scholarly journals A classification of isotropic affine hyperspheres

2016 ◽  
Vol 27 (09) ◽  
pp. 1650074 ◽  
Author(s):  
Marilena Moruz ◽  
Luc Vrancken
Keyword(s):  
Acta Math ◽  
Complete Classification ◽  
Positive Definite ◽  
Affine Hypersurface ◽  
Affine Hypersphere ◽  
Difference Tensor ◽  
Affine Spheres

We study affine hypersurfaces [Formula: see text], which have isotropic difference tensor. Note that, any surface always has isotropic difference tensor. In case that the metric is positive definite, such hypersurfaces have been previously studied in [O. Birembaux and M. Djoric, Isotropic affine spheres, Acta Math. Sinica 28(10) 1955–1972.] and [O. Birembaux and L. Vrancken, Isotropic affine hypersurfaces of dimension 5, J. Math. Anal. Appl. 417(2) (2014) 918–962.] We first show that the dimension of an isotropic affine hypersurface is either [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Next, we assume that [Formula: see text] is an affine hypersphere and we obtain for each of the possible dimensions a complete classification.

A SPECIAL DEGREE REDUCTION OF POLYNOMIALS OVER FINITE FIELDS WITH APPLICATIONS

2011 ◽  
Vol 07 (04) ◽  
pp. 1093-1102 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Explicit Formula ◽  
Finite Fields ◽  
Rational Points ◽  
Degree Reduction ◽  
Affine Hypersurface ◽  
Special Degree ◽  
Low Degree ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a polynomial in n variables over the finite field 𝔽q and Nq(f) denote the number of 𝔽q-rational points on the affine hypersurface f = 0 in 𝔸n(𝔽q). A φ-reduction of f is defined to be a transformation σ : 𝔽q[x1, …, xn] → 𝔽q[x1, …, xn] such that Nq(f) = Nq(σ(f)) and deg f ≥ deg σ(f). In this paper, we investigate φ-reduction by using the degree matrix which is formed by the exponents of the variables of f. With φ-reduction, we may improve various estimates on Nq(f) and utilize the known results for polynomials with low degree. Furthermore, it can be used to find the explicit formula for Nq(f).

The cylinder theorem in $${{\mathcal {H}}}^2\times R$$

2020 ◽  
Vol 111 (3) ◽  
Author(s):  
João Lucas Marques Barbosa ◽  
Manfredo Perdigão do Carmo
Keyword(s):  
Cylinder Theorem

scholarly journals An affine analogue of the Hartman-Nirenberg cylinder theorem

2002 ◽  
Vol 322 (3) ◽  
pp. 573-582
Author(s):  
Maks A. Akivis ◽  
Vladislav V. Goldberg
Keyword(s):  
Cylinder Theorem

RIGIDITY OF AFFINE HYPERSURFACES WITH RANK 1 SHAPE OPERATOR

2003 ◽  
Vol 14 (03) ◽  
pp. 211-234
Author(s):  
LUC VRANCKEN
Keyword(s):  
Curvature Tensor ◽  
Bilinear Form ◽  
Affine Connection ◽  
Shape Operator ◽  
Symmetric Bilinear Form ◽  
Positive Definite ◽  
Affine Hypersurface ◽  
3 Dimensional ◽  
Monge Ampere ◽  
Affine Metric

On a non-degenerate hypersurface it is well known how to induce an affine connection ∇ and a symmetric bilinear form, called the affine metric. Conversely, given a manifold M and an affine connection ∇ one can ask whether this connection is locally realizable as the induced affine connection on a nondegenerate affine hypersurface and to what extend this immersion is unique. In case that the image of the curvature tensor R of ∇ is 2-dimensional and M is at least 3-dimensional a rigidity theorem was obtained in [4]. In this paper, we discuss positive definite n-dimensional affine hypersurfaces with rank 1 shape operator (which is equivalent with 1-dimensional image of the curvature tensor) which are non-rigid. We show how to construct such affine hypersurfaces using solutions of (n - 1)-dimensional differential equations of Monge–Ampère type.

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