arXiv research

Study on hypersurfaces with specific curvature conditions.

Lie minimal surfaces are characterized by differential equations of principal 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 …

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

PCA adapted for curved spaces improves data analysis.

Introduces nonlinear splittings on fibre bundles for generalizing connections.

Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …

This paper develops a Hamiltonian reduction method for field theories over affine principal bundles.

The paper calculates area Siegel--Veech constants for specific submanifolds of REL zero.

Transforms classical connections using pushforwards and gauge transformations.

We prove that holomorphic normal projective connections on compact complex surfaces are flat. We show that a holomorphic torsion-free affine connection $\nabla$ on a compact complex surface is locally modelled on a translations-invariant affine connection on $\C^2$, except if $\nabla$ is a generic connection on a princ…

We study several quantities associated to the Green's function of a multiply connected domain in the complex plane. Among them are some intrinsic properties such as geodesics, curvature, and $L^2$-cohomology of the capacity metric and critical points of the Green's function. The principal idea used is an affine scaling…

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to con…

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left c…

Let $M$ be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric $g$ and a covariant constant volume form. Let $G$ be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat princip…

Develops a unified theory of Yang-Mills and GR using generalized principal bundles.

Classifies holonomy groups of Riemannian manifolds and finds compact ones imply cone structures.

Develops a Kaluza-Klein theory in affine spaces without metric.

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

The paper classifies and determines properties of specific hypersurfaces in complex hyperbolic quadrics.

The paper explores equi-affine curvatures in pseudo-Riemannian manifolds.

Study on isoparametric and constant-curvature hypersurfaces in Finsler spaces.

Holomorphic projective structures and bundles are studied on surfaces, revealing affine spaces of parameters.

Study on discrete surfaces with constant principal curvature for nanocarbon applications.

Study of convex hypersurfaces with specific curvature properties.

Extends PF submanifold results and connects Kac-Moody spaces.

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

Develops radiant structures for statistical manifolds.

Study behavior of curvatures near singular points of frontals.

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

New complete hypersurfaces found in affine geometry.

Classifies surfaces with special curvature properties.

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

Develops a framework for potentials on Lauritzen manifolds using bi-forms.

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $δ\in π_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $δ$ is simply connected. In contrast,…

The paper studies Einstein hypersurfaces in a specific warped product space.

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

In this paper are determined the principal curvatures and principal curvature lines on canal surfaces which are the envelopes of families of spheres with variable radius and centers moving along a closed regular curve in R^3. By means of a connection of the differential equations for these curvature lines and real Ricc…

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

This paper constructs and studies the long-term factorization of affine pricing kernels into discounting at the rate of return on the long bond and the martingale component that accomplishes the change of probability measure to the long forward measure. The principal eigenfunction of the affine pricing kernel germane t…

Abstract mathematical formulas for statistical structures and curvatures.

The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.

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 $t$, …

Reduces field theories on principal bundles by a subgroup, deriving reduced equations.

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.

The paper classifies hypersurfaces with special curvature properties in various spaces.