arXiv research

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this te…

The paper discusses algorithms for reconstructing curves with given Euclidean or affine curvatures.

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines …

Study of convex hypersurfaces with specific curvature properties.

The study classifies certain types of incomplete surfaces with low curvature.

Extends Choi-Wang inequality to Li-Xia affine connections.

In this paper, the Cartan frames and the equi-affine curvatures are described with the help of the Frenet frames and the Frenet curvatures of a non-null and non-degenerate curve in a 3-dimensional pseudo-Riemannian manifold. The constancy of the Frenet curvatures of such a curve always implies the constancy of the equi…

New complete hypersurfaces found in affine geometry.

The paper calculates curvature limits and proves Gauss-Bonnet theorems in affine and Minkowski groups.

We show that every Kaehler affine curvature model can be realized geometrically.

The paper establishes inequalities for convex curves and applies them to lattice point estimates.

Abstract mathematical formulas for statistical structures and curvatures.

At a 3/2-cusp of a given plane curve $γ(t)$, both of the Euclidean curvature $κ_g$ and the affine curvature $κ_A$ diverge. In this paper, we show that each of $\sqrt{|s_g|}κ_g$ and $(s_A)^2 κ_A$ (called the Euclidean and affine normalized curvature, respectively) at a 3/2-cusp is a smooth function of the variable $t$, …

The paper calculates curvature limits and Gauss-Bonnet theorems in the Heisenberg group.

The second boundary value problem of the prescribed affine mean curvature equation is a nonlinear, fourth order, geometric partial differential equation. It was introduced by Trudinger and Wang in 2005 in their investigation of the affine Plateau problem in affine geometry. The previous works of Trudinger-Wang, Chau-We…

Unified description of aesthetic curves through self-affinities.

Comparison theorems in centro-affine differential geometry

Study spherical convex bodies using $L_p$-floating areas and curvature entropy.

After having given the general variational formula for the functionals indicated in the title, the critical points of the integral of the equi-affine curvature under area constraint and the critical points of the full-affine arc-length are studied in greater detail.

We show that every finite-dimensional Alexandrov space X with curvature bounded from below embeds canonically into a product of an Alexandrov space with the same curvature bound and a Euclidean space such that each affine function on X comes from an affine function on the Euclidean space.

Defines CAMC discrete nets and their properties.

For an arbitrary nondegenerate curve in a pseudo-Riemann\-ian (including Riemannian) 2-manifold, we express the equi-affine curvature with the help of the Frenet (geodesic) curvature of this curve.

In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connecti…

Solitons are special polygon midpoints under affine transformations.

The study shows how to foliate convex hypersurfaces in affine space with constant curvature.

The paper classifies affine hypersurfaces with symplectic structures and constraints on their curvature.

The paper proves an inequality and describes a curve flow in centro-affine geometry.

We study a family of 3-dimensional Lorentz manifolds. Some members of the family are 0-curvature homogeneous, 1-affine curvature homogeneous, but not 1-curvature homogeneous. Some are 1-curvature homogeneous but not 2-curvature homogeneous. All are 0-modeled on indecomposible local symmetric spaces. Some of the members…

Study on Kähler manifolds with nonnegative Ricci curvature, focusing on rigidity.

The study finds parametrizations for surfaces of revolution with a linear curvature ratio.

We give characterizations of affine transformations and affine vector fields in terms of the spray. By utilizing the Jacobi type equation that characterizes affine vector fields, we prove some rigidity theorems of affine vector fields on compact or forward complete non-compact Finsler manifolds with non-positive total …

A curvature model (V,A) is a real vector space V which is equipped with a "curvature operator" A(x,y)z that A has the same symmetries as an affine curvature operator; A(x,y)z=-A(y,x)z and A(x,y)z+A(y,z)x+A(z,x)y=0. Such a model is called projective affine Osserman if the spectrum of the Jacobi operator J(y):x->A(x,y)y,…

We use results of Matzeu and Nikcevic to decompose the space of affine Kaehler curvature tensors as a direct sum of irreducible modules in the complex setting

The aim of this paper is to give a local description of affine surfaces, whose induced Blaschke structure is projectively flat. We show that such affine surfaces with constant Gauss affine curvature and indefinite induced Blaschke metric are described by soliton equations.

New method for curvature computation in sub-Riemannian geometry.

This paper finds a metric on \(S^2 imes T^2\) with strictly positive biorthogonal curvature using an affine connection with antisymmetric torsion.

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gau…

Study classifies zero mean curvature surfaces with planar curvature lines.

Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.

In this paper we study the general affine differential geometry of surfaces in affine space $A^3$. For a regular elliptical surface we define a moving frame of minimal order and get the complete system of differential invariants. As an application we classify regular elliptical surfaces of constant curvatures up to aff…

The paper extends affine connection results to singular warped and twisted products.

Symmetry groups help define solitons in curved spaces.

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

The paper classifies curves in dual affine and Lorentz-Minkowski planes with constant curvature.

It is developed the considerations from (S. M. Minčić, [14, 15]) about curvature tensors and pseudotensors for a non-symmetric affine connection space in this paper. How many kinds of covariant derivatives are enough to be defined for complete researching in the field of non-symmetric affine connection spaces is examin…

We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, s…

We construct a sequence of commuting central affine curve flows on $R^n\backslash 0$ invariant under the action of $SL(n,R)$ and prove the following results: (a) The central affine curvatures of a solution of the j-th central affine curve flow is a solution of the j-th flow of Gelfand-Dickey (GD$_n$) hierarchy on the s…

Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.