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searching for Sectional curvature 31 found (121 total)

Thurston elliptization conjecture (216 words) [view diff] exact match in snippet view article find links to article

metric of constant positive sectional curvature. A 3-manifold with a Riemannian metric of constant positive sectional curvature is covered by the 3-sphere

Alexandrov space (275 words) [view diff] exact match in snippet view article find links to article

curvature ≥ k form a generalization of Riemannian manifolds with sectional curvature ≥ k, where k is some real number. By definition, these spaces are

Sharafutdinov's retraction (156 words) [view diff] exact match in snippet view article find links to article

any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric. Perelman later showed that in this setting, Sharafutdinov's

Riemannian submersion (467 words) [view diff] exact match in snippet view article find links to article

lower bound for the sectional curvature of N {\displaystyle N} is at least as big as the lower bound for the sectional curvature of M {\displaystyle M}

Complex hyperbolic space (2,051 words) [view diff] exact match in snippet view article find links to article

being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant

Codazzi tensor (547 words) [view diff] exact match in snippet view article find links to article

sectional curvature, any Codazzi tensor h with trgh constant must be parallel. Furthermore, on a compact manifold of nonnegative sectional curvature,

2π theorem (407 words) [view diff] exact match in snippet view article find links to article

than 2π results in a 3-manifold with a complete metric of negative sectional curvature. In fact, this metric can be selected to be identical to the original

Fubini–Study metric (5,292 words) [view diff] exact match in snippet view article find links to article

The maximum sectional curvature (4) is attained at a holomorphic 2-plane — one for which J(σ) ⊂ σ — while the minimum sectional curvature (1) is attained

Yau's conjecture on the first eigenvalue (410 words) [view diff] exact match in snippet view article find links to article

hypersurface M n {\displaystyle M^{n}} of the standard (n + 1)-sphere with sectional curvature 1 is n {\displaystyle n} If S n + 1 {\displaystyle S^{n+1}} is the

Beltrami's theorem (738 words) [view diff] exact match in snippet view article find links to article

tensor as given above, it follows that the chart domain has constant sectional curvature −1/ngil(∂i ρl − ρi ρl). By connectedness of the manifold, this local

John A. Thorpe (291 words) [view diff] case mismatch in snippet view article find links to article

Columbia University, under the direction of James Eells (Higher Order Sectional Curvature). From 1963 to 1965, he was Moore Instructor at MIT and Assistant

Complex projective plane (500 words) [view diff] exact match in snippet view article find links to article

the complex projective plane is a 4-dimensional manifold whose sectional curvature is quarter-pinched, but not strictly so. That is, it attains both

Volume entropy (626 words) [view diff] exact match in snippet view article find links to article

entropy htop of the geodesic flow on M. Moreover, if M has nonpositive sectional curvature then h = htop. These results are due to Manning. More generally,

Hitchin–Thorpe inequality (919 words) [view diff] exact match in snippet view article find links to article

inequality is obtained, then the Ricci curvature of g is zero; if the sectional curvature is not identically equal to zero, then (M, g) is a Calabi–Yau manifold

Einstein manifold (830 words) [view diff] exact match in snippet view article find links to article

tensor having a single degree of freedom. Any manifold with constant sectional curvature is an Einstein manifold—in particular: Euclidean space, which is

Sphere (5,274 words) [view diff] exact match in snippet view article find links to article

given normal section exists a circle of curvature that equals the sectional curvature, is tangent to the surface, and the center lines of which lie along

Mostow rigidity theorem (1,068 words) [view diff] exact match in snippet view article find links to article

discontinuously (it is equivalent to define it as a Riemannian manifold with sectional curvature -1 which is complete). It is of finite volume if the integral of

Heinz Hopf (957 words) [view diff] exact match in snippet view article find links to article

any simply connected complete Riemannian 3-manifold of constant sectional curvature is globally isometric to Euclidean, spherical, or hyperbolic space

Heegaard splitting (1,949 words) [view diff] exact match in snippet view article find links to article

that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks

Kodaira dimension (2,398 words) [view diff] exact match in snippet view article find links to article

non-positive curvature); see classical theorems, especially on Pinched sectional curvature and Positive curvature. These statements are made more precise below

Yamabe problem (1,418 words) [view diff] no match in snippet view article find links to article

except in the case of the standard sphere with its usual constant-sectional-curvature metric, the only constant-scalar-curvature metrics in the conformal

Laplace–Beltrami operator (3,336 words) [view diff] exact match in snippet view article find links to article

operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded

Laplace–Beltrami operator (3,336 words) [view diff] exact match in snippet view article find links to article

operator on the (n − 1)-sphere with its canonical metric of constant sectional curvature 1. It is convenient to regard the sphere as isometrically embedded

Kobayashi metric (2,246 words) [view diff] exact match in snippet view article find links to article

If a complex manifold X has a Hermitian metric with holomorphic sectional curvature bounded above by a negative constant, then X is Kobayashi hyperbolic

Teichmüller space (4,994 words) [view diff] exact match in snippet view article find links to article

Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic. With the

Geometry Festival (2,867 words) [view diff] exact match in snippet view article find links to article

physics Burkhard Wilking, New examples of manifolds with positive sectional curvature almost everywhere John Roe, Amenability and assembly maps Eleny Ionel

Monochromatic electromagnetic plane wave (3,516 words) [view diff] exact match in snippet view article find links to article

to the direction of propagation) and background (the transverse sectional curvature). In contrast, the Bel decomposition of the Riemann curvature tensor

Arithmetic Fuchsian group (3,844 words) [view diff] exact match in snippet view article find links to article

2-manifold) with a hyperbolic metric (a Riemannian metric of constant sectional curvature −1). If Γ {\displaystyle \Gamma } is an arithmetic Fuchsian group

Forces on sails (8,303 words) [view diff] exact match in snippet view article find links to article

width versus sail height, expressed as an aspect ratio—and cross-sectional curvature or draft. In aerodynamics, the aspect ratio of a sail is the ratio

Chern's conjecture for hypersurfaces in spheres (1,871 words) [view diff] exact match in snippet view article find links to article

University Press (1982), pp. 669–706, problem 105 L. Verstraelen, Sectional curvature of minimal submanifolds, Proceedings of the Workshop on Differential

Computational anatomy (16,865 words) [view diff] case mismatch in snippet view article find links to article

Micheli, Mario; Michor, Peter W.; Mumford, David (2012-03-01). "Sectional Curvature in Terms of the Cometric, with Applications to the Riemannian Manifolds