global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It is called hyperbolic...

manifolds. Causality conditions Globally hyperbolic manifold Hyperbolic partial differential equation Orientable manifold Spacetime Benn & Tucker (1987)...

continuous Causally simple Globally hyperbolic Given are the definitions of these causality conditions for a Lorentzian manifold ( M , g ) {\displaystyle...

model Constructions in hyperbolic geometry Hjelmslev transformation Hyperbolic 3-manifold Hyperbolic manifold Hyperbolic set Hyperbolic tree Kleinian group...

Riemannian manifold. Ricci-flat manifolds are a special kind of Einstein manifold. In theoretical physics, Ricci-flat Lorentzian manifolds are of fundamental...

geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear...

terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential...

Green’s functions of Lorentzian Green hyperbolic 2nd order partial differential equations in a globally hyperbolic manifold, and in the definition of Hadamard...

equations, the stable manifold theorem is an important result about the structure of the set of orbits approaching a given hyperbolic fixed point. It roughly...

gave rise to hyperbolic geometry and elliptic geometry. In the modern theory of manifolds, these notions correspond to Riemannian manifolds with constant...

In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a...

Cauchy surface Closed timelike curve Cosmic censorship hypothesis Globally hyperbolic manifold Malament–Hogarth spacetime Null infinity Penrose diagram Penrose–Hawking...

function-theoretic classification but it is hyperbolic in the geometric classification. Dessin d'enfant Kähler manifold Lorentz surface Mapping class group Serre...

infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood. Those of...

0). Hyperbolic rotation of the plane does not mingle the quadrants, in fact, each one is an invariant set under the unit hyperbola group. 4-manifold Clifford-Klein...

orientable Riemannian 2-manifolds into elliptic/parabolic/hyperbolic cases. Each such manifold has a conformally equivalent Riemannian metric with constant...

manifolds of the dynamical system; they behave physically under small perturbations; and they explain many of the observed statistics of hyperbolic systems...

development is called a globally hyperbolic vacuum development. Choquet-Bruhat also proved a uniqueness theorem: Given any two globally hyperbolic vacuum developments...

globally hyperbolic manifold X = R × M {\displaystyle X=\mathbb {R} \times M} . Since any oriented three-dimensional manifold is parallelizable, a globally hyperbolic...

a pseudo-Riemannian manifold. On a Riemannian manifold it is an elliptic operator, while on a Lorentzian manifold it is hyperbolic. The Laplace–de Rham...

branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent...

is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function...

differentiable manifold, so that the derivative f n ′ {\displaystyle f_{n}^{\prime }} is defined, then one says that a periodic point is hyperbolic if | f n...

feature space or embedding space, is an embedding of a set of items within a manifold in which items resembling each other are positioned closer to one another...

3-manifolds: of the 8 geometries, all but hyperbolic are quite constrained. Dimension 0 is trivial and 1 is straightforward. Low dimension manifolds (dimensions...

n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter...

referred to as hyperbolic LCSs, as they provide a finite-time generalization of the classic concept of normally hyperbolic invariant manifolds in dynamical...

mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. Geometric topology as an...

group of the quaternionic hyperbolic space are arithmetic.[GS92] In 1978, Gromov introduced the notion of almost flat manifolds.[G78] The famous quarter-pinched...

[not verified in body] Suppose M is a compact two-dimensional Riemannian manifold with boundary ∂M. Let K be the Gaussian curvature of M, and let kg be the...