In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion). Let be a Riemannian manifold, and a Riemannian submanifold. Define, for a given , a vector to be normal to whenever for all (so that is orthogonal to ). The set of all such is then called the normal space to at . Just as the total space of the tangent bundle to a manifold is constructed from all tangent spaces to the manifold, the total space of the normal bundle to is defined as The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub-bundle of the cotangent bundle. More abstractly, given an immersion (for instance an embedding), one can define a normal bundle of N in M, by at each point of N, taking the quotient space of the tangent space on M by the tangent space on N. For a Riemannian manifold one can identify this quotient with the orthogonal complement, but in general one cannot (such a choice is equivalent to a of the projection ). Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space restricted to the subspace. Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N: where is the restriction of the tangent bundle on M to N (properly, the pullback of the tangent bundle on M to a vector bundle on N via the map ). The fiber of the normal bundle in is referred to as the normal space at (of in ). If is a smooth submanifold of a manifold , we can pick local coordinates around such that is locally defined by ; then with this choice of coordinates and the ideal sheaf is locally generated by . Therefore we can define a non-degenerate pairing that induces an isomorphism of sheaves . We can rephrase this fact by introducing the conormal bundle defined via the conormal exact sequence then , viz. the sections of the conormal bundle are the cotangent vectors to vanishing on .

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.