About: Sectional curvature

Facets (new session)
Description
Metadata
Settings
- Rule:
- Inverse Functional Properties:
- "Same As":

About: Sectional curvature Goto Sponge NotDistinct Permalink

An Entity of Type : owl:Thing, within Data Space : dbpedia.demo.openlinksw.com associated with source document(s)
QRcode icon

http://dbpedia.demo.openlinksw.com/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FSectional_curvature&invfp=IFP_OFF&sas=SAME_AS_OFF

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

Attributes	Values
rdfs:label	Schnittkrümmung (de) Courbure sectionnelle (fr) Curvatura sezionale (it) 단면 곡률 (ko) Sectiekromming (nl) 断面曲率 (ja) Sectional curvature (en) Секционная кривизна (ru) Секційна кривина (uk) 截面曲率 (zh)
rdfs:comment	Die Schnittkrümmung ist eine Größe der riemannschen Geometrie, eines Teilgebiets der Mathematik. Mit ihrer Hilfe kann man die Krümmung einer -dimensionalen riemannschen Mannigfaltigkeit beschreiben. Dabei wird jeder (zweidimensionalen) Ebene im Tangentialraum an einem Punkt dieser Mannigfaltigkeit eine Zahl als Krümmung zugeordnet. Die Schnittkrümmung kann als Verallgemeinerung der gaußschen Krümmung verstanden werden. Der Name kommt daher, dass man sozusagen einen Schnitt durch die Mannigfaltigkeit in Richtung der gegebenen Ebene legt und die gaußsche Krümmung der so entstandenen Fläche bestimmt. (de) En géométrie riemannienne, la courbure sectionnelle est une des façons de décrire la courbure d'une variété riemannienne. Elle peut être définie à partir du tenseur de courbure, et permet de retrouver ce dernier.On définit une courbure sectionnelle en chaque point et pour chacun des 2-plans inclus dans l'espace tangent à la variété riemannienne en m. Formellement, la collection de toutes les courbures sectionnelles constitue une application sur la grassmannienne des 2-plans, à valeurs réelles. (fr) In geometria differenziale, la curvatura sezionale misura la curvatura di una varietà riemanniana lungo piani dello spazio tangente in un punto della varietà. La curvatura sezionale contiene la stessa quantità di informazioni del tensore di Riemann. (it) 리만 기하학에서 단면 곡률(斷面曲率, 영어: sectional curvature)은 특정한 접평면에 대한 방향으로 리만 다양체가 굽는 양을 나타내는 실수이다. 단면 곡률에 상한 또는 하한을 가하면, 리만 다양체의 다양한 미분기하학·미분위상수학적 정보를 유추할 수 있다. (ko) リーマン幾何学において、断面曲率(英: sectional curvature)は、を記述する方法のひとつである。断面曲率 K(σp) は p の接空間内の 2次元平面 σp に依存する。断面曲率は曲面のガウス曲率であり、σp 方向の点 p から始まる測地線より得られる p での接平面 σp を持つ（言い換えると、この平面は、p でのの下の像である。断面曲率は、多様体上の 2次元のファイバーバンドル上の滑らかな実数値函数である。断面曲率は、リーマン曲率テンソルを完全に決定する。 (ja) Секционная кривизна — один из способов описания кривизны римановых многообразий. (ru) В рімановій геометрії, секційна кривина є однією із кривин ріманового многовиду. Секційна кривина K(σp) залежить від вибору двовимірної площині σp в дотичному просторі в точці p. У двовимірному рімановому многовиді секційна кривина збігається з гаусовою кривиною. Секційна кривина повністю визначається тензором кривини. (uk) 在黎曼几何中，截面曲率是描述的一种方式。截面曲率依赖于p点的切空间裡的一个二维平面。它就定义为该截面，考慮在 p 点以平面作为切平面的曲面，這曲面是收集流形中某包含的鄰域內從 p 点出發的測地線且這測地線在點的切向量屬於截面（換句話說就是其中是里包含原點的鄰域），而截面曲率就是曲面在點的高斯曲率。形式上，截面曲率是流形上的2维格拉斯曼纤维丛的光滑实值函数。截面曲率完全决定了曲率张量，是非常有用的几何概念。 (zh) In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σp as a tangent plane at p, obtained from geodesics which start at p in the directions of σp (in other words, the image of σp under the exponential map at p). The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold. (en) In de Riemann-meetkunde, een deelgebied van de wiskunde, is de sectiekromming een van de manieren om de te beschrijven. De sectiekromming K(σp) is afhankelijk van een twee-dimensionaal vlak σp in de raakruimte op punt p. Het is de Gaussiaanse kromming van het oppervlak dat het vlak σp als een raakvlak in p heeft en dat is verkregen uit geodeten die beginnen in p in de richtingen van σp (in andere woorden, het beeld van σp onder de exponentiële afbeelding in p). De sectiekromming is een gladde reëelwaardige functie op de 2-Grassmanniaan-bundel over de variëteit (nl)
dcterms:subject	Riemannian manifolds Curvature (mathematics) Riemannian geometry
Wikipage page ID	285618 (xsd:integer)
Wikipage revision ID	1117702641 (xsd:integer)
Link from a Wikipage to another Wikipage	Preissman's theorem Princeton University Press Myers theorem Annals of Mathematics Holomorphic sectional curvature Homotopy groups Curvature of Riemannian manifolds Complete space Chelsea Publishing Company Orthonormal Riemannian manifolds Gaussian curvature Geodesic Grassmannian N-sphere Curvature (mathematics) Comparison theorem Élie Cartan Fundamental group Hopf conjecture Springer Publishing Surface (topology) Tangent space Hadamard manifold Curvature Euclidean space Exponential map (Riemannian geometry) Fiber bundle Glossary of Riemannian and metric geometry Graduate Texts in Mathematics Isoperimetric inequality Riemann curvature tensor Riemannian geometry Riemannian manifold Isometry Hyperbolic geometry Hyperbolic space Riemannian geometry Academic Press Diffeomorphic Aspherical space Soul theorem Cartan–Hadamard conjecture Cartan–Hadamard theorem Linearly independent Space form Schur's lemma (Riemannian geometry) Toponogov's theorem Universal cover Tangent vectors Springer-Verlag Synge theorem Distance function Geodesic triangle
Link from a Wikipage to an external page	https://semanticscholar.org/paper/2906e408f5668b32182e60ab0c431483354b0ffa http://projecteuclid.org/euclid.bams/1183530120
sameAs	Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature Sectional curvature
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Cite_book dbt:Cite_encyclopedia dbt:Harv dbt:Refbegin dbt:Refend

Faceted Search & Find service v1.17_git139 as of Feb 29 2024

Alternative Linked Data Documents: ODE Content Formats:

RDF

ODATA

Microdata

About

OpenLink Virtuoso version 08.03.3330 as of Mar 19 2024, on Linux (x86_64-generic-linux-glibc212), Single-Server Edition (378 GB total memory, 67 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2024 OpenLink Software