What Is the Geometry of the Universe? - Quanta Magazine
2020年3月16日 · Only three geometries fit this description: flat, spherical and hyperbolic. Let’s explore these geometries, some topological considerations, and what the cosmological evidence says about which shapes best describe our universe. This is the geometry we learned in school.
Hyperbolic geometry - Wikipedia
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
In this paper we will survey a variety of spectral geometry results that are known to hold in the context of hyperbolic manifolds and discuss the extent to which analogous results hold in the setting of spherical manifolds. We conclude with a number of open problems. 1. Introduction. Let (M, g) be a compact Riemannian manifold.
Here we will look at the basic ideas of hyperbolic geometry including the ideas of lines, distance, angle, angle sum, area and the isometry group and Þnally the construction of Schwartz triangles.
In 1997 Cornell University mathematician Daina Taimina finally worked out how to make a physical model of hyperbolic space that allows us to feel, and to tactilely explore, the properties of this unique geometry. The method she used was crochet.
such as Euclidean, spherical or hyperbolic space, is called a congruence if the distance between every pair of points p and q in one set is the same as the distance between f (p) and f (q) in the other set.
Chapter VIII. Hyperbolic Geometry - MIT OpenCourseWare
There are various ways of drawing the hyperbolic plane in the ordinary Euclidean one; obviously none of them works perfectly. Here we stick with the half-plane model, which is what you are most likely to see where hyperbolic geometry intersects …
One, with positive curvature K, is the spherical space, and the other, with negative curvature K, is the hyperbolic space. We shall not speak again of space with zero curvature, Euclidean space. These three spaces are undoubtedly the simplest Riemannian spaces, the most widely used in …
Spherical Geometry - SpringerLink
2019年10月24日 · Spherical Geometry. In: Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics, vol 149. Springer, Cham. https://doi.org/10.1007/978-3-030-31597-9_2. Anyone you share the following link with will be able to read this content: Provided by the Springer Nature SharedIt content-sharing initiative. Policies and ethics.
[PDF] The spectral geometry of hyperbolic and spherical …
2023年5月18日 · In this paper we will survey a variety of spectral geometry results that are known to hold in the context of hyperbolic manifolds and discuss the extent to which analogous results hold in the setting of spherical manifolds.