In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics...

theorem (2). But the point can be considered to be external to the remaining sphere of radius r, and according to (1) all of the mass of this sphere can...

always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0). The theorem was first proved...

Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if there is a map f : ( D 2 , ∂...

Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used...

disk, the complex plane, or the Riemann sphere. The theorem is a generalization of the Riemann mapping theorem from simply connected open subsets of the...

well. The Dandelin spheres can be used to give elegant modern proofs of two classical theorems known to Apollonius. The first theorem is that a closed conic...

In mathematics, Reeb sphere theorem, named after Georges Reeb, states that A closed oriented connected manifold M n that admits a singular foliation having...

(conjectured by Richard Hamilton). In 2007, he proved the differentiable sphere theorem (in collaboration with Richard Schoen), a fundamental problem in global...

northern hemisphere cut out from a sphere of radius R. Its Euler characteristic is 1. On the left hand side of the theorem, we have K = 1 / R 2 {\displaystyle...

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle...

theorem to spheres, and in another poem described the chain of six spheres each tangent to its neighbors and to three given mutually tangent spheres,...

obtaining a new convergence theorem for Ricci flow. A special case of their convergence theorem has the differentiable sphere theorem as a simple corollary...

A sphere (from Greek σφαῖρα, sphaîra) is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that are all at the...

geometry) Soul theorem (Riemannian geometry) Sphere theorem (Riemannian geometry) Synge's theorem (Riemannian geometry) Toponogov's theorem (Riemannian geometry)...

/ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball...

Papakyriakopoulos in 1956, along with Dehn's lemma and the Sphere theorem. A simple and useful version of the loop theorem states that if for some 3-dimensional manifold...

In mathematics, Kuiper's theorem (after Nicolaas Kuiper) is a result on the topology of operators on an infinite-dimensional, complex Hilbert space H...

suspension theorem, which relates the homotopy groups of a space and its suspension. In the case of spheres, the suspension of an n-sphere is an (n+1)-sphere, and...

of each dimension for a simplicial d-sphere? In the case of polytopal spheres, the answer is given by the g-theorem, proved in 1979 by Billera and Lee (existence)...

cylinder collapsing to its axis, or a sphere collapsing to its center. Perelman's proof of his canonical neighborhoods theorem is a highly technical achievement...

Nash–Kuiper theorem. For example, the image of any smooth isometric hypersurface immersion of the round sphere must itself be a round sphere. By contrast...

resulting in the Jordan–Brouwer separation theorem. Theorem—Let X be an n-dimensional topological sphere in the (n+1)-dimensional Euclidean space Rn+1...

convergence theorem (Brendle & Schoen 2009). Their convergence theorem included as a special case the resolution of the differentiable sphere theorem, which...

hard open question of whether the 4-sphere has non-standard smooth structures. For n = 2, the h-cobordism theorem is equivalent to the Poincaré conjecture...

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f...

Dehn's lemma Loop theorem (aka the Disk theorem) Sphere theorem Haken manifold JSJ decomposition Branched surface Lamination Examples 3-sphere Torus bundles...

the notion of almost flat manifolds.[G78] The famous quarter-pinched sphere theorem in Riemannian geometry says that if a complete Riemannian manifold has...

The Alexander horned sphere is a pathological object in topology discovered by J. W. Alexander (1924). It is a particular topological embedding of a two-dimensional...

In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy...