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Equivariant Bundle
In geometry and topology, given a group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ... ''G'', an equivariant bundle is a fiber bundle such that the total space and the base spaces are both ''G''-spaces and the projection map \pi between them is equivariant: \pi \circ g = g \circ \pi with some extra requirement depending on a typical fiber. For example, an equivariant vector bundle is an equivariant bundle. References *Berline, Nicole; Getzler, E.; Vergne, Michèle (2004), Heat Kernels and Dirac Operators, Berlin, New York: Springer-Verlag Fiber bundles {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group
A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identity * Religious group (other), a group whose members share the same religious identity * Social group, a group whose members share the same social identity * Tribal group, a group whose members share the same tribal identity * Organization, an entity that has a collective goal and is linked to an external environment * Peer group, an entity of three or more people with similar age, ability, experience, and interest Social science * In-group and out-group * Primary, secondary, and reference groups * Social group * Collectives Science and technology Mathematics * Group (mathematics), a set together with a binary operation satisfying certain algebraic conditions Chemistry * Functional group, a group of atoms which provide s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group Action
In 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__ Definition and first properties Let \sigma: G \times M \to M, (g, x) \mapsto g \cdot x 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 \sigma is differentiable. Equivalently, a Lie group action of G on M consists of a Lie group homomorphism G \to \mathrm(M). A smooth manifold endowed with a Lie group action is also called a ''G''-manifold. The fact that the action map \sigma is smooth has a couple of immediate consequences: * the stabilizers G_x \subseteq G of the group action are closed, thus are Lie subgroups of ''G'' * the orbits G \cdot x \subseteq M of the group action are immersed submanifolds. Forgetting the smooth structure, a Lie group action is a particular case of a continuous group action. Examples For ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Vector Bundle
In mathematics, given an action \sigma: G \times_S X \to X of a group scheme ''G'' on a scheme ''X'' over a base scheme ''S'', an equivariant sheaf ''F'' on ''X'' is a sheaf of \mathcal_X-modules together with the isomorphism of \mathcal_-modules :\phi: \sigma^* F \xrightarrow p_2^*F that satisfies the cocycle condition: writing ''m'' for multiplication, :p_^* \phi \circ (1_G \times \sigma)^* \phi = (m \times 1_X)^* \phi. Notes on the definition On the stalk level, the cocycle condition says that the isomorphism F_ \simeq F_x is the same as the composition F_ \simeq F_ \simeq F_x; i.e., the associativity of the group action. The condition that the unit of the group acts as the identity is also a consequence: apply (e \times e \times 1)^*, e: S \to G to both sides to get (e \times 1)^* \phi \circ (e \times 1)^* \phi = (e \times 1)^* \phi and so (e \times 1)^* \phi is the identity. Note that \phi is an additional data; it is "a lift" of the action of ''G'' on ''X'' to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
