We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piecewise linear isotropic maps can be chosen so that they are piecewise linear isotropic immersions as well. The proofs are obtained using analogies with an infinite dimensional moment map geometry due to Donaldson. As a byproduct of these considerations, we introduce a numerical flow in finite dimension, whose limit provide, from an experimental perspective, many examples of piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely available, is based on the Euler method and shows the evolution equation of discrete surfaces in real time, as a movie.
Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.
We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.
Let $\mathbb{Q}_3$ be the complex 3-quadric endowed with its standard complex conformal structure. We study the complex conformal geometry of isotropic curves in $\mathbb{Q}_3$. By an isotropic curve we mean a nonconstant holomorphic map from a Riemann surface into $\mathbb{Q}_3$, null with respect to the conformal structure of $\mathbb{Q}_3$. The relations between isotropic curves and a number of relevant classes of surfaces in Riemannian and Lorentzian spaceforms are discussed.
We study invariant surfaces generated by one-parameter subgroups of simply and pseudo isotropic rigid motions. Basically, the simply and pseudo isotropic geometries are the study of a three-dimensional space equipped with a rank 2 metric of index zero and one, respectively. We show that the one-parameter subgroups of isotropic rigid motions lead to seven types of invariant surfaces, which then generalizes the study of revolution and helicoidal surfaces in Euclidean and Lorentzian spaces to the context of singular metrics. After computing the two fundamental forms of these surfaces and their Ga…
A production function is a mathematical formalization in economics which denotes the relations between the output generated by a firm, an industry or an economy and the inputs that have been used in obtaining it. In this paper, we study the product production functions of 2 variables in terms of the geometry of their associated graph surfaces in the isotropic 3-space I^{3}. In particular, we derive several classification results for the graph surfaces of product production functions in I^{3} with constant curvature.
In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is based on previous results in the simply isotropic space [B. Pavković, Glas. Mat. Ser. III $\mathbf{15}$, 149 (1980); Rad JAZU $\mathbf{450}$, 129 (1990)], which point to the possibility of introducing an isotropic Gauss map taking values on a unit sphere of parabolic type and of…
We give a construction to obtain canonically an ``isotropic average'' of given $C^1$-close isotropic submanifolds of a symplectic manifold. To do so we use an improvement of Weinstein's submanifold averaging theorem (obtained in collaboration with H. Karcher) and apply ``Moser's trick''. We also present an application to Hamiltonian group actions.
Unlocking Euclidean Problems with Isotropic Initialization
King Abdullah University of Science and Technology（阿卜杜拉国王科技大学）；MAIS, Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；RWTH Aachen University（亚琛工业大学）；TU Wien（维也纳科技大学）；Xi’an Jiaotong University（西安交通大学）
This paper investigates the hanging chain problem in the simply isotropic plane as well as its 2-dimensional analog in the simply isotropic space. The simply isotropic plane and space are two- and three-dimensional geometries equipped with a degenerate metric whose kernel has dimension 1. Although the metric is degenerate, the hanging chain and hanging surface problems are well-posed if we employ the relative arc length and relative area to measure the weight. Here, the concepts of relative arc length and relative area emerge by seeing the simply isotropic geometry as a relative geometry. In a…
Zero mean curvature surfaces in the simply isotropic 3-space $\mathbb{I}^3$ naturally appear as intermediate geometry between geometry of minimal surfaces in $\mathbb{E}^3$ and that of maximal surfaces in $\mathbb{L}^3$. In this paper, we investigate reflection principles for zero mean curvature surfaces in $\mathbb{I}^3$ as with the above surfaces in $\mathbb{E}^3$ and $\mathbb{L}^3$. In particular, we show a reflection principle for isotropic line segments on such zero mean curvature surfaces in $\mathbb{I}^3$, along which the induced metrics become singular.
Simply-connected homogeneous spacetimes for kinematical and aristotelian Lie algebras (with space isotropy) have recently been classified in all dimensions. In this paper, we continue the study of these "maximally symmetric" spacetimes by investigating their local geometry. For each such spacetime and relative to exponential coordinates, we calculate the (infinitesimal) action of the kinematical symmetries, paying particular attention to the action of the boosts, showing in almost all cases that they act with generic non-compact orbits. We also calculate the soldering form, the associated viel…
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discret…
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of even basic concepts such as discrete curvature κ, discrete torsion τ, or discrete Frenet frame.
In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to c…
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a certain convexity condition have at least d+1 flattenings. This result provides a new approach to the above mentioned classical theorems.
We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $Γ$-convergence. Variants of this discrete energy have been discussed before in the computer graphics literature.
The conformal geometry of the Schwarzian Davey-Stewartson II hierarchy and its discrete analogue is investigated. Connections with discrete and continuous isothermic surfaces and generalised Clifford configurations are recorded. An interpretation of the Schwarzian Davey-Stewartson II flows as integrable deformations of conformally immersed surfaces is given.
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic line bundles over "discrete surfaces", by which we mean the vertex sets of triangulated surfaces with bi-colored set of faces. The resulting theory of discrete conformality is simultaneously Moebius invariant and based on linear equations. In the special case of maps into the 2-sp…
In this paper we study invariant almost Hermitian geometry on generalized flag manifolds which the isotropy representation decompose into two or three irreducible components. We will provide a classification of such flag manifolds admitting Kähler like scalar curvature metric, that is, almost Hermitian structures $(g,J)$ satisfying $s=2s_C$ where $s$ is Riemannian scalar curvature and $s_C$ is the Chern scalar curvature.
Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature…
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discre…
It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible. In particular result can be applied directly to known classes of almost (para-) contact metric manifolds when considered Jacobi operator of characteristic vector field has maximal rank. There is effective algorithmic procedure which resolves the problem of existence of such vect…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer…
In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a brief review of foundations of nonassociative geometry,we discuss the nonassociative smooth and discrete de Sitter spacetimes.
For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of ``nonelementary'' discrete isometry groups of higher rank symmetric spaces. We obtain…
The asymptotic lattices and their transformations are studied within the line geometry approach. It is shown that the discrete asymptotic nets are represented by isotropic congruences in the Plucker quadric. On the basis of the Lelieuvre-type representation of asymptotic lattices and of the discrete analog of the Moutard transformation, it is constructed the discrete analog of the W-congruences, which provide the Darboux-Backlund type transformation of asymptotic lattices.The permutability theorems for the discrete Moutard transformation and for the corresponding transformation of asymptotic l…
