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- In [[Riemannian geometry]], '''Schur's lemma''' is a result that says, heuristically, whenever certa Suppose <math>(M, g)</math> is a smooth [[Riemannian manifold]] with dimension <math>n.</math> Recall that this defines for each ...14 KB (2,268 words) - 16:56, 17 October 2024
- {{short description|Isometry group of a compact Riemannian manifold with negative Ricci curvature is finite}} ...Bochner]] proved in 1946 that any [[Killing vector field]] of a compact [[Riemannian manifold]] with negative [[Ricci curvature]] must be zero. Consequently the ...6 KB (849 words) - 10:10, 19 April 2022
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- {{Short description|Type of Riemannian metric}} ...ric''' is a natural choice of Riemannian metric on the tangent bundle of a Riemannian manifold. ...1 KB (225 words) - 05:35, 8 February 2024
- {{Short description|Triangle comparison theorem in Riemannian geometry}} In the [[mathematics|mathematical]] field of [[Riemannian geometry]], '''Toponogov's theorem''' (named after [[Victor Andreevich Toponogov]]) ...2 KB (339 words) - 01:27, 12 August 2023
- {{about|the exponential map in differential geometry|discrete dynamical systems|Exponential map (discrete dynamical systems)}} In [[differential geometry]], the '''exponential map''' is a generalization of the ordinary [[exponent ...848 bytes (118 words) - 00:24, 14 December 2017
- In [[spectral geometry]], the '''Rayleigh–Faber–Krahn inequality''', named after its conjecturer, ...ok|url=http://worldcat.org/oclc/1106800772|title=Eigenvalues in Riemannian geometry|last=Chavel|first= Isaac|date=1984|oclc=1106800772}}</ref> In particular, a ...2 KB (249 words) - 17:53, 22 December 2024
- ...fold''' is an [[almost-contact manifold]] endowed with a certain kind of [[Riemannian metric]]. They are named after the Japanese mathematician Katsuei Kenmotsu. ...hi, \xi, \eta)</math> be an [[almost-contact manifold]]. One says that a [[Riemannian metric]] <math>g</math> on <math>M</math> is adapted to the almost-contact ...2 KB (339 words) - 14:17, 4 October 2024
- {{Short description|The isometry group of a Riemannian manifold is a Lie group}} ...[[Smooth function|smooth]] [[Isometry (Riemannian geometry)|isometry]] of Riemannian manifolds. A simpler proof was subsequently given by [[Richard Palais]] in ...3 KB (374 words) - 01:18, 25 December 2024
- ...Mikhail Gromov]].<ref>{{cite journal |last=Gromov |first=M. |title=Filling Riemannian manifolds |journal=J. Diff. Geom. |volume=18 |date=1983 |pages=1–147 |cites * [[Systolic geometry]] ...2 KB (290 words) - 20:09, 8 January 2025
- In mathematics, the '''Besicovitch inequality''' is a [[Geometry|geometric inequality]] relating volume of a set and distances between certa ...r the n-dimensional cube <math>[0,1]^n</math> with a [[Riemannian geometry|Riemannian]] metric <math>g</math>. Let ...3 KB (350 words) - 03:56, 20 September 2024
- ...connection coincides with the [[Levi-Civita connection]] of the associated Riemannian metric. * Shoshichi Kobayashi, ''Differential geometry of complex vector bundles''. Publications of the Mathematical Society of Ja ...2 KB (265 words) - 16:27, 4 February 2025
- ...a''') is a fundamental equation in the analysis of [[spinor]]s on [[pseudo-Riemannian manifold]]s. In dimension 4, it forms a piece of [[Seiberg–Witten gauge th Given a [[spin structure]] on a pseudo-Riemannian manifold ''M'' and a [[spinor bundle]] ''S'', the Lichnerowicz formula stat ...3 KB (443 words) - 15:56, 12 December 2024
- ...ecially in connection with the [[filling area conjecture]] in [[Riemannian geometry]],{{r|katz}} but this term has also been used for other concepts.{{r|kurita ...distance between the same points on a [[sphere]], and on the [[hemisphere (geometry)|hemisphere]]s into which the circle divides the sphere.]] ...5 KB (652 words) - 04:21, 1 July 2024
- ...low]]. It is of considerable interest in differential geometry to find the Riemannian metric on a given [[smooth manifold]] which minimizes the volume entropy, w Let (''M'', ''g'') be a compact Riemannian manifold, with [[universal cover]] <math>\tilde{M}.</math> Choose a point < ...4 KB (624 words) - 11:25, 13 May 2021
- ...Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds | last1 = Chanu | first1= Claudia | last2 = Rastelli|first2 = Gio ...(\mathcal S^1,\dots,\mathcal S^n)</math> of ''n'' pairwise [[transversal (geometry)|transversal]] and orthogonal [[foliation]]s of connected [[submanifold]]s ...4 KB (541 words) - 10:18, 19 April 2022
- ...parabolic geometry in this sense is also called a parabolic geometry: any geometry that is modeled on such a space by means of a [[Cartan connection]]. ...o the tangent spaces of the base manifold. Broadly speaking, [[projective geometry]] refers to the study of manifolds with this kind of connection. ...3 KB (480 words) - 22:27, 10 January 2024
- In [[differential geometry]], a '''Clifford module bundle''', a '''bundle of Clifford modules''' or ju Given an oriented [[Riemannian manifold]] ''M'' one can ask whether it is possible to construct a bundle o ...3 KB (415 words) - 18:44, 29 January 2024
- {{Short description|Derivative in differential geometry and vector calculus}} In the math branches of [[differential geometry]] and [[vector calculus]], the '''second [[covariant derivative]]''', or th ...3 KB (528 words) - 04:28, 26 June 2024
- In [[geometry]], if ''X'' is a manifold with an action of a [[topological group]] ''G'' b === Riemannian examples === ...8 KB (1,354 words) - 16:05, 24 January 2025
- ...ries.'' In: Springer LNM, 1620 (1996), pp. 120–348.</ref> is a flat [[Riemannian manifold]] with a certain compatible multiplicative structure on the [[tang ...lly [[quantum cohomology]]. The broadest definition is in the category of Riemannian [[supermanifold]]s. We will limit the discussion here to smooth (real) man ...4 KB (652 words) - 16:18, 10 January 2025
- ...ntegral of [[Gaussian curvature]] of a non-compact [[surface (differential geometry)|surface]] to the [[Euler characteristic]]. It is akin to the [[Gauss–Bonn ...c on the manifold. Cohn-Vossen's inequality states that in every complete Riemannian 2-manifold ''S'' with finite [[total curvature]] and finite Euler character ...4 KB (498 words) - 09:48, 14 August 2023
- {{short description|Type of Riemannian manifold with constant Jacobi operator spectrum}} ...the volume of the intersection of two geodesic balls |journal=Differential Geometry and Its Applications}}</ref> It is named after [[Americans|American]] [[mat ...5 KB (727 words) - 22:08, 5 February 2025