About: Chern–Gauss–Bonnet theorem

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

About: Chern–Gauss–Bonnet theorem Goto Sponge NotDistinct Permalink

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

http://dbpedia.org/describe/?url=http%3A%2F%2Fdbpedia.org%2Fresource%2FChern%E2%80%93Gauss%E2%80%93Bonnet_theorem

In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler-Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an ). Riemann-Roch and Atiyah-Singer are other generalizations of the Gauss-Bonnet theorem.

Attributes	Values
rdfs:label	Satz von Chern-Gauß-Bonnet (de) Teorema de Gauss-Bonnet generalizado (es) Chern–Gauss–Bonnet theorem (en) 一般ガウス・ボネの定理 (ja) Gegeneraliseerde stelling van Gauss-Bonnet (nl) Обобщённая формула Гаусса — Бонне (ru) 陳-高斯-博內定理 (zh)
rdfs:comment	En matemáticas, el teorema de Gauss-Bonnet generalizado presenta la característica de Euler de una variedad de Riemann cerrada como integral de cierto polinomio derivado de su curvatura. Es una generalización directa del teorema de Gauss-Bonnet a la dimensión par en general. (es) 陈・ボネの定理（Chern–Gauss–Bonnet theorem，チャーン・ガウス・ボネの定理とも呼ばれる）は、偶数次元の閉リーマン多様体のオイラー特性数を曲率から導かれるある多項式の積分として表す定理である。 M を境界のないコンパクトな向き付け可能な 2n 次元リーマン多様体とし、Ω をレヴィ・チヴィタ接続の曲率形式とする。これは、Ω が M 上のに値を持つ 2-形式であることを意味する。そのために、Ω は成分が 2-形式である反対称 2n × 2n 行列であるので、可換環上の行列である。従って、2n-形式を成分にもつパフィアン Pf(Ω) をとることができる。この状況で一般ガウス・ボネの定理はとなる。ここで χ(M) は、M のオイラー数を表す。この定理は、ガウス・ボネの定理の高次元化である。 (ja) In de differentiaalmeetkunde, een deelgebied van de wiskunde, is de gegeneraliseerde stelling van Gauss-Bonnet (ook wel de stelling van Chern-Gauss-Bonnet genoemd) een stelling die zegt dat de euler-karakteristiek van een gesloten, even-dimensionale riemann-variëteit gelijk is aan de integraal van een bepaalde polynoom van haar kromming. Het is een directe generalisatie van de stelling van Gauss-Bonnet (genoemd naar Carl Friedrich Gauss en Pierre Ossian Bonnet) naar hogere dimensies die voor het eerst gepubliceerd werd door Shiing-Shen Chern in 1945, en daarmee een verband legde tussen globale topologie en lokale meetkunde. (nl) Обобщенная формула Гаусса — Бонне — интегральная формула, выражающая эйлерову характеристику замкнутого чётномерного риманова многообразия через его кривизну. Это прямое обобщение формулы Гаусса — Бонне на высшие размерности. (ru) 在數學中，陳定理（或陳–高斯–博內定理）以数学家陈省身、卡尔·弗里德里克·高斯、皮埃尔·奥西恩·博内（页面存档备份，存于互联网档案馆）的名字命名。此定理断言：2n維黎曼流形的歐拉示性數可以從曲率計算出來。陳定理也是高斯–博內定理（n=1）在高维的推廣，其在數學和理論物理學中亦有许多應用。此定理由陈省身於1945年證出。陳定理將全局拓扑學與局部微分几何联系起來。 (zh) In mathematics, the Chern theorem (or the Chern–Gauss–Bonnet theorem after Shiing-Shen Chern, Carl Friedrich Gauss, and Pierre Ossian Bonnet) states that the Euler-Poincaré characteristic (a topological invariant defined as the alternating sum of the Betti numbers of a topological space) of a closed even-dimensional Riemannian manifold is equal to the integral of a certain polynomial (the Euler class) of its curvature form (an ). Riemann-Roch and Atiyah-Singer are other generalizations of the Gauss-Bonnet theorem. (en)
dcterms:subject	Theorems in differential geometry
Wikipage page ID	393115 (xsd:integer)
Wikipage revision ID	1093826787 (xsd:integer)
Link from a Wikipage to another Wikipage	Carl B. Allendoerfer Carl Friedrich Gauss Princeton University Scalar curvature Metric connection Volume element Euclidean field theory Euler-Poincaré characteristic Topological phases of matter De Rham cohomology Betti number Ricci curvature Characteristic class Curvature form Curvature of Riemannian manifolds Invariant polynomial Topological invariant Compact space Mathematics Chern–Simons form Chern–Simons theory Chern–Weil homomorphism Gauss map Geometric phase Tangent bundle Gauss–Bonnet theorem General relativity Geometry N-sphere Condensed matter physics André Weil Levi-Civita connection String theory Closed manifold Commutative ring Pfaffian Physics Topology K-theory Pontryagin class Duncan Haldane Elliptic equation Isomorphism Riemann curvature tensor Riemannian manifold Riemannian metric Heinz Hopf Atiyah–Singer index theorem Theorems in differential geometry Jacobian matrix and determinant Hypersurface Chern class Surface (mathematics) Manifold Pierre Ossian Bonnet Positive-definite Special orthogonal group Institute for Advanced Study Integral Nash embedding theorem Orbifold Orientability Cap product Shiing-Shen Chern Euler characteristic Euler class Symbol of a differential operator Supersymmetry Characteristic classes Topological quantum field theory Pontryagin number Riemann–Roch theorem S.-S. Chern 2-form Elliptic differential operator Naturality Adjoint operator Cohomology class Berry's phase Boundary (of a manifold) dbr:Analytical_invariant dbr:Gauss-Kronecker_curvature
sameAs	Chern–Gauss–Bonnet theorem Chern–Gauss–Bonnet theorem

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, 59 GB memory in use)
Data on this page belongs to its respective rights holders.
Virtuoso Faceted Browser Copyright © 2009-2024 OpenLink Software