vector bundle

The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold it its own right. The dimension of T(M) is twice the dimension of M. Each tangent space of an n-dimensional vector space is an n-dimensional vector space. As a set then, T(M) is isomorphic to M × Rn. As a manifold, however, T(M) is not always diffeomorphic to the product manifold M à ...

See also:

Tangent bundle, Tangent bundle - Topology and smooth structure, Tangent bundle - Examples, Tangent bundle - Vector fields

Read more here: » Tangent bundle: Encyclopedia II - Tangent bundle - Topology and smooth structure

A principal G-bundle is a fiber bundle Ï€ : P → X together with a continuous right action P × G → P by a topological group G such that G preserves the fibers of P and acts freely and transitively on them. (One often requires the base space X to be a Hausdorff space and possibly paracompact). The abstract fiber of the bundle is taken to be G itself. It follows that the orbits of the G-action are precisely the fibers of Ï€ : P → < ...

See also:

Principal bundle, Principal bundle - Formal definition, Principal bundle - Examples, Principal bundle - Characterization of principal bundles, Principal bundle - Reduction of the structure group

Read more here: » Principal bundle: Encyclopedia II - Principal bundle - Formal definition

A fiber bundle consists of the data (E, B, Ï€, F), where E, B, and F are topological spaces and Ï€ : E → B is a continuous surjection satisfying a local triviality condition outlined below. B is called the base space of the bundle, E the total space, and F the fiber. The map Ï€ is called the projection map. We shall assume in what ...

See also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

Read more here: » Fiber bundle: Encyclopedia II - Fiber bundle - Formal definition

In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX-modules OXm → OXn. Here OX is the structure sheaf of local rings, given by definition on X. The form of the definition is a global (on X) way of carrying across the idea of a finitely-presented mo ...

Including:

• Coherent sheaf - Coherent cohomology

Read more here: » Coherent sheaf: Encyclopedia - Coherent sheaf

A smooth assignment of a vector at each point of a manifold is called a vector field. Specifically, a vector field on a manifold M is a smooth map such that the image of x, denoted Vx, lies in Tx(M), the tangent space to x. In the language of fiber bundles, such a map is called a section. A vector field on M is theref ...

See also:

Tangent bundle, Tangent bundle - Topology and smooth structure, Tangent bundle - Examples, Tangent bundle - Vector fields

Read more here: » Tangent bundle: Encyclopedia II - Tangent bundle - Vector fields

In mathematics, an involution, or an involutary function, is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. Involution - General properties. The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other exa ...

Including:

• Involution - General properties
• Involution - Involutions in Euclidean geometry
• Involution - Involutions in differential geometry
• Involution - Involutions in ring theory
• Involution - Involutions in group theory

Read more here: » Involution: Encyclopedia - Involution

In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way. More precisely, a complex manifold has an atlas of charts to Cn, such that the change of coordinates between charts are holomorphic. Complex manifolds can be regarded as a special case of differentiable manifolds. For example, a 1-dimensional complex manifold is geometrically a surface, known as a Riemann surface. The requirement that the transition functions b ...

Including:

• Complex manifold - Examples of complex manifolds
• Complex manifold - Integrable almost-complex structures
• Complex manifold - Kähler and Calabi-Yau manifolds

Read more here: » Complex manifold: Encyclopedia - Complex manifold

André Weil (May 6, 1906 - August 6, 1998) was one of the great mathematicians of the 20th century. He is known for his foundational work in number theory and algebraic geometry. He was a founding member, and de facto the early leader, of the influential Bourbaki group. The philosopher Simone Weil was his sister. André Weil - Life. Born in Paris to Alsatian parents who fled the annexation of Alsace-Lorraine to Germany, he studied in Paris, Rome and Göttingen and received his doctorate in 1928. He s ...

Including:

• André Weil - Life
• André Weil - Work
• André Weil - Books
• André Weil - External link
• André Weil - Bibliography

Read more here: » André Weil: Encyclopedia - André Weil

The simplest example is that of Rn. In this case the tangent bundle is trivial and isomorphic to R2n. Another simple example is the unit circle, S1. The tangent bundle is of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder of infinite height. Unfortunately, the only tangent bundles that can be readily visualized are those of the real line R and the unit circle S1, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dim ...

See also:

Tangent bundle, Tangent bundle - Topology and smooth structure, Tangent bundle - Examples, Tangent bundle - Vector fields

Read more here: » Tangent bundle: Encyclopedia II - Tangent bundle - Examples

Given a subgroup , one may consider the bundle P / H whose fibers are the cosets G / H. If the new bundle admits a global section, then one says that the section is a reduction of the structure group from G to H . In particular, if H is the identity, then a section of P itself is a reduction of the structure group to the ...

See also:

Principal bundle, Principal bundle - Formal definition, Principal bundle - Examples, Principal bundle - Characterization of principal bundles, Principal bundle - Reduction of the structure group

Read more here: » Principal bundle: Encyclopedia II - Principal bundle - Reduction of the structure group

A section (or cross section) of a fiber bundle is a continuous map f : B → E such that Ï€(f(x))=x for all x in B. Since bundles do not in general have globally-defined sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology. Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle ...

See also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

Read more here: » Fiber bundle: Encyclopedia II - Fiber bundle - Sections

Let E = B × F and let Ï€ : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle. Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. The Möbius strip has a circle for a base B and a line segment for the fiber F. A neighborhood U of a point is an arc; in the picture, this is th ...

See also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

Read more here: » Fiber bundle: Encyclopedia II - Fiber bundle - Examples

The most common example of a smooth principal bundle is the frame bundle of a smooth manifold M. Here the fiber over a point x in M is the set of all frames (i.e. ordered bases) for the tangent space TxM. The general linear group GL(n,R) acts simply-transitively on these frames. These fibers can be glued together in a natural way so as to obtain a principa ...

See also:

Principal bundle, Principal bundle - Formal definition, Principal bundle - Examples, Principal bundle - Characterization of principal bundles, Principal bundle - Reduction of the structure group

Read more here: » Principal bundle: Encyclopedia II - Principal bundle - Examples

Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (USee also:

Fiber bundle, Fiber bundle - Formal definition, Fiber bundle - Examples, Fiber bundle - Sections, Fiber bundle - Structure groups and transition functions

Read more here: » Fiber bundle: Encyclopedia II - Fiber bundle - Structure groups and transition functions

Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariant constructions, in the way that a section is a kind of function on a space, and to lead to a contradiction from the existence of a section we do need that variance. In fact cohomology theory grew up after homology and homotopy theory, which are both covariant theories based on mapping into a space; and characteristic class theory in its infancy in the 1930s (as part of obstruction theory) was one major reason wh ...

See also:

Characteristic class, Characteristic class - Definition, Characteristic class - Motivation

Read more here: » Characteristic class: Encyclopedia II - Characteristic class - Motivation

At its roots, geometry consists of a notion of "congruence" between different objects in a space. In the late 19th century, notions of congruence were typically supplied by the action of a Lie group on space. Lie groups generally act quite rigidly, and so a Cartan geometry is a generalization of this notion of congruence to allow for curvature to be present. Of course, a flat Cartan geometry should be a geometry without curvature. Beginning then with the flat case, we describe what is meant by a Cartan Geometry in general formal mathematical terms. ...

See also:

Cartan connection, Cartan connection - Conceptual aspects of the theory, Cartan connection - A general theory of frames, Cartan connection - Identifying the tangent bundle, Cartan connection - General theory in formal terms, Cartan connection - The flat case, Cartan connection - The curved case, Cartan connection - Gauges for a Cartan connection, Cartan connection - The fundamental D operator, Cartan connection - Covariant differentiation

Read more here: » Cartan connection: Encyclopedia II - Cartan connection - General theory in formal terms

Assume from now on all mappings are continuous functions from a topological space to another. One says that p : E → B has the homotopy lifting property with respect to a space X if for any homotopy g : X × [0,1] → B and map h : X → E such that p o h = g|X × 0 there is a homotopy f : X × [0,1] → E ...

See also:

Homotopy lifting property, Homotopy lifting property - Formal definition, Homotopy lifting property - Generalizations

Read more here: » Homotopy lifting property: Encyclopedia II - Homotopy lifting property - Formal definition

Let E be a rank k vector bundle over a smooth manifold M and let ∇ be a connection on E. Given a piecewise smooth loop γ : [0,1] → M based at x in M, the connection defines a parallel transport map . This map is both linear and invertible and so defines an element of GL(Ex). The holonomy group of ∇ based at x is defined as The local holonomy group based at x is th ...

See also:

Holonomy, Holonomy - On vector bundles, Holonomy - Riemannian holonomy groups, Holonomy - Special holonomy manifolds in string theory, Holonomy - On principal bundles, Holonomy - Reference, Holonomy - External link

Read more here: » Holonomy: Encyclopedia II - Holonomy - On vector bundles

The cotangent bundle X=T*M, since it is a vector bundle, can be regarded as a manifold in its own right. Because of the manner in which the definition of T*M relates to the differential topology of the base space M, X possess a canonical one-form θ (also tautological one-form or symplectic potential). The exterior derivative of θ is a symplectic 2-form, out of which a non-degenerate volume form can be built for X. For example, as a result X is always an orientable manifold (mea ...

See also:

Cotangent bundle, Cotangent bundle - One-forms the cotangent sheaf, Cotangent bundle - Definition of the cotangent sheaf, Cotangent bundle - The cotangent bundle as phase space, Cotangent bundle - The canonical one-form, Cotangent bundle - Symplectic form, Cotangent bundle - Phase space

Read more here: » Cotangent bundle: Encyclopedia II - Cotangent bundle - The cotangent bundle as phase space

If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an equivariant fashion. Jets do not transform as tensors. In fact, jets of functions between two manifolds belong to a Jet bundle. This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a vector bundle, analog ...

See also:

Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between two manifolds