Related papers: Integral-Geometric Formulas for Perimeter in S^2, …

This is a review of the relation between supersymmetric non-linear sigma models and target space geometry. In particular, we report on the derivation of generalized K\"ahler geometry from sigma models with additional spinorial superfields.…

In this paper, we establish a Heintze-Karcher type inequality for hypersurfaces with capillary boundary of contact angle $\theta\in (0,\frac{\pi}{2})$ in a half space or a half ball, by using solution to a mixed boundary value problem in…

This article describes some aspects of Cauchy integrals and related geometry of sets and measures in Euclidean spaces, etc.

We prove the following result on the timelike spherical Hilbert geometry of simplices: Let $\Delta_2$ be a simplex on the 2-sphere and $\tilde{\Delta}_2$ the antipodal simplex. We show that the timelike spherical Hilbert geometry associated…

In this paper, we establish a Heintze-Karcher-type inequality for capillary hypersurfaces in a unit ball. To achieve this, we introduce a special Finsler metric given by Zermelo's navigation and study the geodesic normal flow with respect…

Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\frac{1}{2}g^{ij}p_{i}p_{j}$ are the geodesics. Given a symplectic manifold (\Gamma,\omega), a hamiltonian $H:\Gamma\to\mathbb{R}$ and a…

It is shown that gravity on the line can be described by the two dimensional (2D) Hilbert-Einstein Lagrangian supplemented by a kinetic term for the coframe and a translational {\it boundary} term. The resulting model is equivalent to a…

We use the intrinsic area to define a distance on the space of homothety classes of convex bodies in the $n$-dimensional Euclidean space, which makes it isometric to a convex subset of the infinite dimensional hyperbolic space. The ambient…

We investigate the special K\"ahler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special K\"ahler metric about any point in the base may be computed…

We consider plane symmetric gravitational fields within the framework of General Relativity in (D+1)-dimensional spacetime. Two classes of vacuum solutions correspond to higher-dimensional generalizations of the Rindler and Taub spacetimes.…

We consider polygonal tilings of certain regions and use these to give intuitive definitions of tiling-based perimeter and area. We apply these definitions to rhombic tilings of Elnitsky polygons, computing sharp bounds and average values…

We show the analogy of Cauchy's surface area formula for the Heisenberg groups $\mathbb{H}_n$ for $n\geq 1$, which states that the p-area of any compact hypersurface $\Sigma$ in $\mathbb{H}_n$ with its p-normal vector defined almost…

We propose a generalization of two classes of Lie-Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie-Hamilton systems on Riemannian spaces (flat $2$-torus, product of hyperbolic lines, sphere and…

Let $m\in\mathbb{N},$ $m\geq 2,$ and let $\{p_j\}_{j=1}^m$ be a finite subset of $\mathbb{S}^2$ such that $0\in\mathbb{R}^3$ lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the…

Michor and Mumford have shown that the distances between planar curves in the simplest metric (not involving derivatives) are identically zero. We consider two conformally equivalent metrics for which the distances between curves are…

Generalized complex geometry is a new mathematical framework that is useful for describing the target space of N=(2,2) nonlinear sigma-models. The most direct relation is obtained at the N=(1,1) level when the sigma model is formulated with…

Using the standard Cayley transform and elementary tools it is reiterated that the conformal compactification of the Minkowski space involves not only the "cone at infinity" but also the 2-sphere that is at the base of this cone. We…

It is shown that, classically, the W-algebras are directly related to the extrinsic geometry of the embedding of two-dimensional manifolds with chiral parametrisation (W-surfaces) into higher dimensional K\"ahler manifolds. We study the…

The Gaussian curvature of a two-dimensional Riemannian manifold is uniquely determined by the choice of the metric. The formulas for computing the curvature in terms of components of the metric, in isothermal coordinates, involve the…

The current work focuses on the Gaussian-Minkowski problem in dimension 2. In particular, we show that if the Gaussian surface area measure is proportional to the spherical Lebesgue measure, then the corresponding convex body has to be a…