Complex Curves — Origins and Intrinsic Geometry

1994 ◽  
pp. 39-50
Author(s):  
J. J. Gray
Keyword(s):  
Intrinsic Geometry ◽  
Complex Curves

scholarly journals Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones

2017 ◽  
Vol 4 (1) ◽  
pp. 43-72 ◽  
Author(s):  
Martin de Borbon
Keyword(s):  
Vector Field ◽  
Einstein Metrics ◽  
Projective Line ◽  
Tangent Cones ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
Complex Curves ◽  
Complex Dimensions ◽  
Irregular Case ◽  
Kähler Einstein Metrics

Abstract The goal of this article is to provide a construction and classification, in the case of two complex dimensions, of the possible tangent cones at points of limit spaces of non-collapsed sequences of Kähler-Einstein metrics with cone singularities. The proofs and constructions are completely elementary, nevertheless they have an intrinsic beauty. In a few words; tangent cones correspond to spherical metrics with cone singularities in the projective line by means of the Kähler quotient construction with respect to the S1-action generated by the Reeb vector field, except in the irregular case ℂβ₁×ℂβ₂ with β₂/ β₁ ∉ Q.

scholarly journals Coarse median algebras: the intrinsic geometry of coarse median spaces and their intervals

2021 ◽  
Vol 27 (2) ◽  
Author(s):  
Graham A. Niblo ◽  
Nick Wright ◽  
Jiawen Zhang
Keyword(s):  
Finite Rank ◽  
Intrinsic Geometry ◽  
Bounded Geometry ◽  
Higher Dimensional ◽  
Median Algebra ◽  
Asymptotic Geometry ◽  
Median Operator

AbstractThis paper establishes a new combinatorial framework for the study of coarse median spaces, bridging the worlds of asymptotic geometry, algebra and combinatorics. We introduce a simple and entirely algebraic notion of coarse median algebra which simultaneously generalises the concepts of bounded geometry coarse median spaces and classical discrete median algebras. We study the coarse median universe from the perspective of intervals, with a particular focus on cardinality as a proxy for distance. In particular we prove that the metric on a quasi-geodesic coarse median space of bounded geometry can be constructed up to quasi-isometry using only the coarse median operator. Finally we develop a concept of rank for coarse median algebras in terms of the geometry of intervals and show that the notion of finite rank coarse median algebra provides a natural higher dimensional analogue of Gromov’s concept of $$\delta $$ δ -hyperbolicity.

Lecture on SUSY Gauge Theories and Integrable Systems

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3093-3124
Author(s):  
A. Marshakov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integrable Systems ◽  
Gauge Theories ◽  
Toda Chain ◽  
Quantum Field ◽  
Standard Quantum ◽  
Integrable Equations ◽  
Complex Curves ◽  
Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

The Intrinsic Geometry of the Cerebral Cortex

1994 ◽  
Vol 166 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Lewis D. Griffin
Keyword(s):  
Cerebral Cortex ◽  
Intrinsic Geometry

Intrinsic geometry of Lax equation

2003 ◽  
Vol 6 (3) ◽  
pp. 291-299
Author(s):  
Carlo Cattani ◽  
Ettore Laserra
Keyword(s):  
Intrinsic Geometry ◽  
Lax Equation

Complex Curves as Lines of Geometries

2016 ◽  
Vol 71 (1-2) ◽  
pp. 145-165 ◽  
Author(s):  
Olga Belova ◽  
Josef Mikeš ◽  
Karl Strambach
Keyword(s):  
Complex Curves

scholarly journals KNOT SINGULARITIES OF HARMONIC MORPHISMS

2001 ◽  
Vol 44 (1) ◽  
pp. 71-85 ◽  
Author(s):  
Paul Baird
Keyword(s):  
Riemann Surface ◽  
Complex Surface ◽  
Subject Classification ◽  
Harmonic Morphisms ◽  
Singular Set ◽  
Harmonic Morphism ◽  
Complex Curves ◽  
Analytic Curve ◽  
Primary 57M25 ◽  
Straight Lines

AbstractA harmonic morphism defined on $\mathbb{R}^3$ with values in a Riemann surface is characterized in terms of a complex analytic curve in the complex surface of straight lines. We show how, to a certain family of complex curves, the singular set of the corresponding harmonic morphism has an isolated component consisting of a continuously embedded knot.AMS 2000 Mathematics subject classification: Primary 57M25. Secondary 57M12; 58E20

Shape Synthesis and Optimization Using Intrinsic Geometry

1990 ◽  
Author(s):  
Shahriar Tavakkoli ◽  
Sanjay G. Dhande
Keyword(s):  
Optimization Problems ◽  
Piecewise Linear ◽  
Shape Design ◽  
Arc Length ◽  
Intrinsic Geometry ◽  
Shape Model ◽  
Design Variables ◽  
Dimensional Shape ◽  
Variable Geometry Truss ◽  
Arc Lengths

Abstract The present paper outlines a method of shape synthesis using intrinsic geometry to be used for two-dimensional shape optimization problems. It is observed that the shape of a curve can be defined in terms of intrinsic parameters such as the curvature as a function of the arc length. The method of shape synthesis, proposed here, consists of selecting a shape model, defining a set of shape design variables and then evaluating Cartesian coordinates of a curve. A shape model is conceived as a set of continuous piecewise linear segments of the curvature; each segment defined as a function of the arc length. The shape design variables are the values of curvature and/or arc lengths at some of the end-points of the linear segments. The proposed method of shape synthesis and optimization is general in nature. It has been shown how the proposed method can be used to find the optimal shape of a planar Variable Geometry Truss (VGT) manipulator for a pre-specified position and orientation of the end-effector. In conclusion, it can be said that the proposed approach requires fewer design variables as compared to the methods where shape is represented using spline-like functions.

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