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...

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

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 , ∂...

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...

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

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...

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

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...

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

manifold or on the behavior of points at "sufficiently large" distances. Sphere theorem. If M is a simply connected compact n-dimensional Riemannian manifold...

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

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...

Euclid does not include the area and volume of a sphere, only a theorem that the volume of a sphere varies as the third power of its diameter, probably...

of the sphere and of a point mass located in the sphere's center. This theorem thus refers to the first statement of Isaac Newton’s shell theorem (the identity...

theory) Sperner's theorem (combinatorics) Sphere theorem (Riemannian geometry) Spin–statistics theorem (physics) Sprague–Grundy theorem (combinatorial game...

curvature in any given conformal class. In the case of the Riemann sphere, the Gauss–Bonnet theorem implies that a constant-curvature metric must have positive...

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

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,...

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

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...

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

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...

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...

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...

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

compact Riemann surface is a complex algebraic curve by Chow's theorem and the Riemann–Roch theorem. There are several equivalent definitions of a Riemann surface...

would look like if the theorem were true. To that end, suppose the yellow line in Figure 1 goes through the center of the sphere and is the axis of rotation...

attraction is in equilibrium inside such a sphere (per the shell theorem), other means such as rotating the sphere have to be employed in order to keep things...