Dual spaceIn mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space.
Lie theoryIn mathematics, the mathematician Sophus Lie (liː ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformation groups, which is one of the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence.
Produit directLa plupart des structures algébriques permettent de construire de façon très simple une structure produit sur le produit cartésien des ensembles sous-jacents. Plus généralement, . C'est le cas de la topologie produit dans la catégorie des espaces topologiques. Soient E un ensemble muni d'une loi de composition interne et F un ensemble muni d'une loi de composition interne . On peut définir une loi de composition interne sur le produit cartésien E×F de la façon suivante : Si et sont associatives, alors la loi est associative.
Lie group actionIn differential geometry, a Lie group action is a group action adapted to the smooth setting: G is a Lie group, M is a smooth manifold, and the action map is differentiable. TOC Let be a (left) group action of a Lie group G on a smooth manifold M; it is called a Lie group action (or smooth action) if the map is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism . A smooth manifold endowed with a Lie group action is also called a G-manifold.
Cartan connectionIn the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (repère mobile).
Dual bundleIn mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. The dual bundle of a vector bundle is the vector bundle whose fibers are the dual spaces to the fibers of . Equivalently, can be defined as the Hom bundle that is, the vector bundle of morphisms from to the trivial line bundle Given a local trivialization of with transition functions a local trivialization of is given by the same open cover of with transition functions (the inverse of the transpose).
Espace vectoriel quotientEn algèbre linéaire, l'espace vectoriel quotient E/F d'un espace vectoriel E par un sous-espace vectoriel F est la structure naturelle d'espace vectoriel sur l'ensemble quotient de E par la relation d'équivalence définie de la manière suivante : v est en relation avec w si et seulement si v – w appartient à F. C'est donc l'ensemble des classes [v] = v + F, où v parcourt E, muni des lois suivantes : somme vectorielle : [v] + [w] = [v + w] ; multiplication par un scalaire : λ [v] = [λ v].
Smooth schemeIn algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. First, let X be an affine scheme of finite type over a field k. Equivalently, X has a closed immersion into affine space An over k for some natural number n.
Smooth structureIn mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold. A smooth structure on a manifold is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold is an atlas for such that each transition function is a smooth map, and two smooth atlases for are smoothly equivalent provided their union is again a smooth atlas for This gives a natural equivalence relation on the set of smooth atlases.
Canonical bundleIn mathematics, the canonical bundle of a non-singular algebraic variety of dimension over a field is the line bundle , which is the nth exterior power of the cotangent bundle on . Over the complex numbers, it is the determinant bundle of the holomorphic cotangent bundle . Equivalently, it is the line bundle of holomorphic n-forms on . This is the dualising object for Serre duality on . It may equally well be considered as an invertible sheaf.
