For the concept in homotopy, see collapse (topology).
In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as i goes to infinity the manifold is close to a k-dimensional space, where k < n, in the Gromov–Hausdorff distance sense. Generally there are some restrictions on the sectional curvatures of (M, gi). The simplest example is a flat manifold, whose metric can be rescaled by 1/i, so that the manifold is close to a point, but its curvature remains 0 for all i.
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In Riemannian geometry, a collapsing or collapsedmanifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as...
concept CollapsingmanifoldCollapse, the action of collapsing or telescoping objects Collapsing user interface elements Accordion (GUI) -- collapsing list...
diameter going to zero, so the almost-flat manifold is a special case of a collapsingmanifold, which is collapsing along all directions. According to the...
manifolds while keeping their curvature bounded, II, with Mikhail Gromov. Journal of Differential Geometry. 31, 4. 1990. 269–298. Collapsingmanifold...
Christoffel symbol Collapsingmanifold Complete space Completion Conformal map is a map which preserves angles. Conformally flat a manifold M is conformally...
where g is a continuous function from B0 into A0. Perelman’s collapsing theorem for 3-manifolds, the capstone of his proof of Thurston's geometrization conjecture...
gave a talk entitled, Collapsing Riemannian Manifolds and its Applications. Fukaya, Kenji (1987). "Collapsing Riemannian Manifolds and Eigenvalues of Laplace-Operators"...
renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifoldcollapses to a point in finite time.) However, this doesn't prove the full geometrization...
temperature is no longer high enough to prevent it from collapsing under its own weight. The collapse may be stopped by the degeneracy pressure of the star's...
air directly into the atmosphere. A vacuum hose connected to the intake manifold downstream of the throttle provides a reference pressure for the blowoff...
spacetime manifold depicted by a Lorentzian manifold, and Euclidean wormholes (named after Euclidean manifold, a structure of Riemannian manifold). The Casimir...
In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold...
aspect of a smooth manifold's topology. This diffeomorphism invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may...
theorems embed any abstract manifold in some Euclidean space. The Nash embedding theorem embeds an abstract Riemannian manifold isometrically in a Euclidean...
involves the Universe shaped like a 3-dimensional sphere (or another compact manifold) until a rolling scalar field begins inflating it. These models are notable...
geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics...
Park 10 miles north of Ashbourne, Derbyshire. A natural gateway to the Manifold Valley, the river crosses the road from Hartington (the nearest village)...
× 77.4 mm (2.93 in × 3.05 in) 9.6:1 compression ratio In 1997 the intake manifold was changed again along with the ECU and the result was 85 PS (63 kW; 84 hp)...
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