Solder form | EPFL Graph Search

Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?

Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur GraphSearch.

Découvrez-en plus sur les applications Graph.

In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitively, soldering expresses in abstract terms the idea that a manifold may have a point of contact with a certain model Klein geometry at each point. In extrinsic differential geometry, the soldering is simply expressed by the tangency of the model space to the manifold. In intrinsic geometry, other techniques are needed to express it. Soldering was introduced in this general form by Charles Ehresmann in 1950. Let M be a smooth manifold, and G a Lie group, and let E be a smooth fibre bundle over M with structure group G. Suppose that G acts transitively on the typical fibre F of E, and that dim F = dim M. A soldering of E to M consists of the following data: A distinguished section o : M → E. A linear isomorphism of vector bundles θ : TM → o*VE from the tangent bundle of M to the pullback of the vertical bundle of E along the distinguished section. In particular, this latter condition can be interpreted as saying that θ determines a linear isomorphism from the tangent space of M at x to the (vertical) tangent space of the fibre at the point determined by the distinguished section. The form θ is called the solder form for the soldering. By convention, whenever the choice of soldering is unique or canonically determined, the solder form is called the canonical form, or the tautological form. Suppose that E is an affine vector bundle (a vector bundle without a choice of zero section). Then a soldering on E specifies first a distinguished section: that is, a choice of zero section o, so that E may be identified as a vector bundle. The solder form is then a linear isomorphism However, for a vector bundle there is a canonical isomorphism between the vertical space at the origin and the fibre VoE ≈ E.

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.

Publications associées (1)

Concepts associés (12)

Séances de cours associées (1)

Tetrad formalism

The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called a tetrad or vierbein. It is a special case of the more general idea of a vielbein formalism, which is set in (pseudo-)Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to (pseudo-)Riemannian manifolds in general, and even to spin manifolds.

G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form.

Vector-valued differential form

In mathematics, a vector-valued differential form on a manifold M is a differential form on M with values in a vector space V. More generally, it is a differential form with values in some vector bundle E over M. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Let M be a smooth manifold and E → M be a smooth vector bundle over M.

Dispensing and hermetic sealing Rb in a miniature reference cell for integrated atomic clocks

Herbert Shea, Peter Ryser, Thomas Maeder, Fabrizio Vecchio, Vinu Lalgudi Venkatraman

An innovative method for filling and sealing the reference cell for a Rubidium (Rb) integrated atomic clock is presented. We used a simple, low-temperature solder sealing technique for producing mini-cells of the size of 12 x 10 x 2 mm, suitable for Rb min ...

Elsevier Science, Reg Sales Off, Customer Support Dept, 655 Ave Of The Americas, New York, Ny 10010 Usa2010

Tenseurs et transformations de Lorentz PHYS-427: Relativity and cosmology I

Couvre les tenseurs, les transformations de Lorentz et l'électrodynamique en notation relativiste.