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In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry.

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rdfs:label	Structure spinorielle (fr) Struttura di spin (it) 스핀 다양체 (ko) スピン構造 (ja) Spin structure (en)
rdfs:comment	In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry. (en) 미분위상수학에서 스핀 다양체(spin多樣體, 영어: spin manifold)는 스피너장을 정의할 수 있는 다양체다. 즉, 직교 틀다발 을 이중 피복 공간 에 대하여 적절히 주다발 으로 확장할 수 있는 가향 (준) 리만 다양체다. (ko) 微分幾何学において、向き付け可能リーマン多様体 (M, g) 上のスピン構造（スピンこうぞう、英: spin structure）は、付随するの定義を可能にし、微分幾何学におけるスピノルの概念を生じる。数理物理学、特に場の量子論へ広く応用され、電荷を持たないフェルミオンに関する任意の理論の定義にスピン構造は必須である。純粋数学的にも、微分幾何学や代数的位相幾何学、K-理論などに於いてスピン構造は興味の対象である。スピン構造はに対する基礎付けを成す。 (ja) En géométrie différentielle, il est possible de définir sur certaines variétés riemanniennes la notion de structure spinorielle (qui se décline en structures Spin ou Spinc), étendant ainsi les considérations algébriques sur le groupe spinoriel et les spineurs. En termes imagés, il s'agit de trouver, dans le cadre des « espaces courbes », une géométrie « cachée » à l’œuvre derrière les concepts géométriques ordinaires. On peut aussi y voir une généralisation de la notion d'orientabilité et de changement d'orientation à une forme d'« orientabilité d'ordre supérieur ». Comme l'orientabilité, la présence de structures spinorielles n'est pas universelle mais se heurte à des obstructions qui peuvent être formulées en termes de classes caractéristiques. (fr) In matematica, e in particolare in geometria differenziale, una struttura di spin definita su una varietà riemanniana orientabile (M, g) consente di definire i fibrati spinoriali associati, dando origine alla nozione di campo spinoriale. (it)
dcterms:subject	Structures on manifolds Riemannian manifolds K-theory Algebraic topology Mathematical physics
Wikipage page ID	2844303 (xsd:integer)
Wikipage revision ID	1113944806 (xsd:integer)
Link from a Wikipage to another Wikipage	Princeton University Press Quantum field theory Boundary condition Metaplectic structure Algebraic topology Almost complex manifold Homomorphism Riemann surface Universal coefficient theorem Stiefel–Whitney class Structures on manifolds Max Karoubi Genus (mathematics) Normal bundle Obstruction theory Tangent bundle Riemannian manifolds Edward Witten Friedrich Hirzebruch Geometry Multiplication N-skeleton André Haefliger Long exact sequence Bockstein homomorphism Calabi–Yau manifold Short exact sequence String theory Complex manifold dbr:Complex_projective_plane Complex projective space Complex vector bundle Â genus Orthonormal basis Particle physics Partition function (quantum field theory) Spin group Mathematical physics K-theory Central extension (mathematics) D-branes Affine space American Mathematical Society 3-manifold Algebraic topology Fermions Fiber bundle Elliptic equation Frame bundle Equivariant Orthonormal frame Riemannian manifold Atiyah–Singer index theorem Hurewicz theorem Field theory (physics) Armand Borel Mathematical physics Chern class Supergravity Triangulation (topology) Differential geometry Arrow CW complex Special orthogonal group Spin representation Spinor Fermion

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