Topic
Hyperbolic triangle
About: Hyperbolic triangle is a research topic. Over the lifetime, 1001 publications have been published within this topic receiving 15240 citations.
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Papers
TL;DR: In the case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in a surface is isometric to a neighborhood in the hyper-bolic plane, and the surface itself is the quotient of the hyperbola by a discrete group of motions as discussed by the authors.
Abstract: 1. A conjectural picture of 3-manifolds. A major thrust of mathematics in the late 19th century, in which Poincare had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Three-manifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3-manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the
1,672 citations
Book•
17 Apr 1981
TL;DR: In this article, the authors present a tract on the initial value problem for hyperbolic equations with multiple points having multiple points and define a fundamental solution for the Cauchy problem.
Abstract: The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction...1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation ...7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations...15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence ...2 9 The wave equation ...32 The initial value problem for hyperbolic equations with a normal surface having multiple points ...36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution ...43 The Cauchy problem ...45 Solution of the inhomogeneous equation with a plane wave function as right hand side ...49 The fundamental solution...
651 citations
Book•
1 Apr 1987
TL;DR: In this article, the negative Curvature on Line Bundles (NCLB) is used to define the normal families of the Disc in Pn Minus Hyperplanes.
Abstract: 0 Preliminaries.- I Basic Properties.- II Hyperbolic Imbeddings.- III Brody's Theorem.- IV Negative Curvature on Line Bundles.- V Curvature on Vector Bundles.- VI Nevanlinna Theory.- VII Applications to Holomorphic Curves in Pn.- VIII Normal Families of the Disc in Pn Minus Hyperplanes.
369 citations
TL;DR: The Hyperbolic Number Plane (HPN) as discussed by the authors is a hyperbolic number plane that is a special case of the number plane of the HPN, and it can be represented as follows
Abstract: (1995). The Hyperbolic Number Plane. The College Mathematics Journal: Vol. 26, No. 4, pp. 268-280.
222 citations
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| Year | Papers |
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| 2025 | 6 |
| 2024 | 13 |
| 2023 | 17 |
| 2022 | 37 |
| 2021 | 13 |
| 2020 | 5 |