vector bundle - Article Index

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Topology And Smooth Structure
The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a ...   » Read the article

Encyclopedia - Principal Bundle: Encyclopedia Ii - Principal Bundle - Formal Definition
A principal G-bundle is a fiber bundle π : P → X together with a continuous right action P × G → P by a topological group G su...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Formal Definition
A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surject...   » Read the article

Encyclopedia - Coherent Sheaf: Encyclopedia - Coherent Sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a...   » Read the article

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Vector Fields
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 s...   » Read the article

Encyclopedia - Involution: Encyclopedia - Involution
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 do...   » Read the article

Encyclopedia - Complex Manifold: Encyclopedia - Complex Manifold
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space in a coherent way....   » Read the article

Encyclopedia - André Weil: Encyclopedia - André Weil
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...   » Read the article

Encyclopedia - Tangent Bundle: Encyclopedia Ii - Tangent Bundle - Examples
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 ...   » Read the article

Encyclopedia - Principal Bundle: Encyclopedia Ii - Principal Bundle - Reduction Of The Structure Group
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 o...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Sections
A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Examples
Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just loc...   » Read the article

Encyclopedia - Principal Bundle: Encyclopedia Ii - Principal 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 th...   » Read the article

Encyclopedia - Fiber Bundle: Encyclopedia Ii - Fiber Bundle - Structure Groups And Transition Functions
Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization chart...   » Read the article

Encyclopedia - Characteristic Class: Encyclopedia Ii - Characteristic Class - Motivation
Characteristic classes are in an essential way phenomena of cohomology theory — they are contravariant constructions, in the way that a...   » Read the article

Encyclopedia - Cartan Connection: Encyclopedia Ii - Cartan Connection - General Theory In Formal Terms
At its roots, geometry consists of a notion of "congruence" between different objects in a space. In the late 19th century, notions of co...   » Read the article

Encyclopedia - Homotopy Lifting Property: Encyclopedia Ii - Homotopy Lifting Property - Formal Definition
Assume from now on all mappings are continuous functions from a topological space to another. One says that p : E → B has the ho...   » Read the article

Encyclopedia - Holonomy: Encyclopedia Ii - Holonomy - On Vector Bundles
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] ...   » Read the article

Encyclopedia - Cotangent Bundle: Encyclopedia Ii - Cotangent Bundle - The Cotangent Bundle As Phase Space
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 t...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Functions Between Two Manifolds
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 loca...   » Read the article

Encyclopedia - Connection Mathematics: Encyclopedia Ii - Connection Mathematics - General Concept
The general concept can be summarized as follows: given a fiber bundle with E the total space and B the base space, the tangent space ...   » Read the article

Encyclopedia - Connection Form: Encyclopedia Ii - Connection Form - Principal Bundles
For a principal G-bundle , for each let Tx(E) denote the tangent space at x and Vx the vertical subspace tangent to the fiber . Then con...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - What 'contravariant' Means
Contravariant is a mathematical term with a precise definition in tensor analysis. It specifies precisely the method (direction of projec...   » Read the article

Encyclopedia - Covariant Derivative: Encyclopedia Ii - Covariant Derivative - General Concept
The covariant derivative (also written as ) of a vector u in the direction of the vector v is a rule that defines a third vector called ...   » Read the article

Encyclopedia - Hirzebruch-riemann-roch Theorem: Encyclopedia Ii - Hirzebruch-riemann-roch Theorem - Statement Of Hirzebruch-riemann-roch Theorem
The Hirzebruch-Riemann-Roch theorem applies to any holomorphic vector bundle E on a compact complex manifold X, to calculate the holomorp...   » Read the article

Encyclopedia - Dolbeault Cohomology: Encyclopedia Ii - Dolbeault Cohomology - Construction Of The Cohomology Groups
Let Ωp,q be the vector bundle of complex differential forms of degree (p,q). In the article on complex forms, the Dolbeault operator is ...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Properties Of Chern Classes
Given a complex vector bundle V over a topological space X, the Chern classes of V are a sequence of elements of the cohomology of X. The...   » Read the article

Encyclopedia - Almost Complex Manifold: Encyclopedia Ii - Almost Complex Manifold - Integrable Almost Complex Structures
Every complex manifold is itself an almost complex manifold. In local holomorphic coordinates zμ = xμ + iyμ one can define the maps ...   » Read the article

Encyclopedia - Riemannian Manifold: Encyclopedia Ii - Riemannian Manifold - Introduction
The tangent bundle of a smooth manifold M (or indeed, any vector bundle over a manifold) is, at a fixed point, just a vector space and ea...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By A Triangulation
Orientability, for surfaces, is easily defined, regardless of whether the surface is embedded in an ambient space or not. Any surface has...   » Read the article

Encyclopedia - André Weil: Encyclopedia Ii - André Weil - Work
He made substantial contributions in many areas, the most important being profound connections between algebraic geometry and number theo...   » Read the article

Encyclopedia - Tensor Field: Encyclopedia Ii - Tensor Field - The C∞m Module Explanation
There is another more abstract (but often useful) way of characterizing tensor fields on a manifold M which turns out to actually make te...   » Read the article

Encyclopedia - Poincaré Half-plane Model: Encyclopedia Ii - Poincaré Half-plane Model - Symmetry Groups
A variety of different groups appear in the discussion of the upper half-plane. One is the linear group GL(2,C), called the Möbius group...   » Read the article

Encyclopedia - Involution: Encyclopedia Ii - Involution - General Properties
The identity map is a trivial example of an involution. Common examples in mathematics of more interesting involutions include multiplica...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - An Example On The Circle
We start by considering complex-valued functions on the circle that are "square integrable"(i.e., elements of L2) and have no Fourier coe...   » Read the article

Encyclopedia - Cotangent Bundle: Encyclopedia Ii - Cotangent Bundle - One-forms The Cotangent Sheaf
Smooth sections of the cotangent bundle are differential one-forms. Cotangent bundle - Definition of the cotangent sheaf. Let M×M be t...   » Read the article

Encyclopedia - Covariant Derivative: Encyclopedia Ii - Covariant Derivative - Notation
In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation. Often a notatio...   » Read the article

Encyclopedia - Covariant Derivative: Encyclopedia Ii - Covariant Derivative - Coordinate Description
Given coordinate functions , any tangent vector can be described by its components in the basis . The covariant derivative is a vector an...   » Read the article

Encyclopedia - Covariant Derivative: Encyclopedia Ii - Covariant Derivative - Derivative Along Curve
Since the covariant derivative depends only on value of X at the point one can define covarient derivative along a smooth curve γ(t) in...   » Read the article

Encyclopedia - Tensor Field: Encyclopedia Ii - Tensor Field - Twisting By A Line Bundle
An extension of the tensor field idea incorporates an extra line bundle L on M. If W is the tensor product bundle of V with L, then W is ...   » Read the article

Encyclopedia - Covariant Derivative: Encyclopedia Ii - Covariant Derivative - Formal Definition
Covariant derivative - Functions. Given a function f, the covariant derivative coincides with the normal differentiation of a real fun...   » Read the article

Encyclopedia - Tensor Field: Encyclopedia Ii - Tensor Field - Notation
The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle TM = T(M)...   » Read the article

Encyclopedia - Tensor Field: Encyclopedia Ii - Tensor Field - Geometric Introduction
The geometric intuition for a vector field is of an 'arrow' attached to each point of a region, with variable length and direction. Our i...   » Read the article

Encyclopedia - Connection Form: Encyclopedia Ii - Connection Form - Vector Bundles
The connection form for the vector bundle is the form on the total space of the associated principal bundle, but it can also be completel...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Functions Between Euclidean Spaces
Before giving a rigorous definition of a jet, it is useful to examine some special cases. Jet mathematics - Example: One-dimensional cas...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets At A Point In Euclidean Space: Rigorous Definitions
This subsection focuses on two different rigorous definitions of the jet of a function at a point, followed by a discussion of Taylor's t...   » Read the article

Encyclopedia - Holonomy: Encyclopedia Ii - Holonomy - On Principal Bundles
The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let P be a principal G-bundle over a smooth...   » Read the article

Encyclopedia - Poincaré Half-plane Model: Encyclopedia Ii - Poincaré Half-plane Model - Isometric Symmetry
The group action of the special linear group PSL(2,R) on H is defined by Note that the action is transitive, in that for any , there e...   » Read the article

Encyclopedia - Jet Mathematics: Encyclopedia Ii - Jet Mathematics - Jets Of Sections
This subsection deals with the notion of jets of local sections a vector bundle. Almost everything in this section generalizes mutatis mu...   » Read the article

Encyclopedia - Hirzebruch-riemann-roch Theorem: Encyclopedia Ii - Hirzebruch-riemann-roch Theorem - Riemann Roch Theorem For Curves
For curves, the Hirzebruch-Riemann-Roch theorem is essentially the classical Riemann-Roch theorem. To see this, recall that for each divi...   » Read the article

Encyclopedia - Poincaré Half-plane Model: Encyclopedia Ii - Poincaré Half-plane Model - Geodesics
The geodesics for this metric tensor are circular arcs perpendicular to the real axis (half-circles whose origin is on the real axis) and...   » Read the article

Encyclopedia - Tensor Field: Encyclopedia Ii - Tensor Field - The Vector Bundle Explanation
The contemporary mathematical expression of the idea of tensor field breaks it down into a two-step concept. There is the idea of vector ...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Example: The Complex Tangent Bundle Of The Riemann Sphere
Let CP1 be the Riemann sphere: 1-dimensional complex projective space. Suppose that z is a holomorphic local coordinate for the Riemann s...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - More Formal Statement
We start with a compact smooth manifold M (without boundary), a vector bundle, "E" on M and an elliptic operator D on M. Here "D" is a di...   » Read the article

Encyclopedia - Almost Complex Manifold: Encyclopedia Ii - Almost Complex Manifold - Differential Topology Of Almost Complex Manifolds
Just as a complex structure on a vector space V allows a decomposition of VC into V+ and V-, so an almost complex structure on M allows a...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - History
The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to t...   » Read the article

Encyclopedia - Atiyah–singer Index Theorem: Encyclopedia Ii - Atiyah–singer Index Theorem - Proof Techniques
The index theorem has been proved, and reproved, as a general statement. Atiyah-Singer comment that the initial proof was based on that o...   » Read the article

Encyclopedia - Cartan Connection: Encyclopedia Ii - Cartan Connection - Conceptual Aspects Of The Theory
It was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames. It operates with differential forms ...   » Read the article

Encyclopedia - Almost Complex Manifold: Encyclopedia Ii - Almost Complex Manifold - Formal Definition
Let M be a smooth manifold. An almost complex structure J on M is a linear complex structure (that is, a linear map which squares to −1...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Orientation By Top-dimensional Forms
Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at each point in th...   » Read the article

Encyclopedia - Involution: Encyclopedia Ii - Involution - Involutions In Euclidean Geometry
A simple example of an involution of the three-dimensional Euclidean space is reflection against a plane. Doing a reflection twice, bring...   » Read the article

Encyclopedia - Involution: Encyclopedia Ii - Involution - Involutions In Ring Theory
In ring theory, the word involution is customarily taken to mean an antihomomorphism that is its own inverse function. Examples include c...   » Read the article

Encyclopedia - André Weil: Encyclopedia Ii - 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 rec...   » Read the article

Encyclopedia - Orientability: Encyclopedia Ii - Orientability - Examples In Low Dimensions
Surfaces we normally encounter in every day life are orientable. For example, sphere, plane, torus. Example of non-orientable surfaces ar...   » Read the article

Encyclopedia - Cartan Connection: Encyclopedia Ii - Cartan Connection - Gauges For A Cartan Connection
In performing actual calculations with a Cartan connection, it is traditional to work in a particular gauge. A gauge on M is just a -valu...   » Read the article

Encyclopedia - Cartan Connection: Encyclopedia Ii - Cartan Connection - The Fundamental D Operator
Let V be a real or complex representation of H, with the action of H denoted by ρ. Let A0(P,V) be the space of equivariant V-valued func...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - The Chern Character
The Chern classes can be used to construct a homomorphism of rings from the topological K-theory of a space to (the completion of) its ra...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Chern Numbers
If we work on an oriented manifold of dimension 2n, then any product of Chern classes of total degree 2n can be paired with the orientati...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Informal Usage
In common physics usage, the adjective covariant may sometimes be used informally as a synonym for invariant (or equivariant, in mathemat...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Example: Covariant Basis Vectors In Euclidean R3
If e1, e2, e3 are contravariant basis vectors of R3 (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of the...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Usage In Tensor Analysis
In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths,...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Chern Classes Of Almost Complex Manifolds And Cobordism
The theory of Chern classes gives rise to cobordism invariants for almost complex manifolds. If M is an almost complex manifold, then its...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Chern Classes Of Line Bundles
If V is a line bundle there is just a single (first) Chern class in the second cohomology group of X. The first Chern class turns out to...   » Read the article

Encyclopedia - Characteristic Class: Encyclopedia Ii - Characteristic Class - Definition
Let G be a group, and for a topological space X, write bG(X) for the set of isomorphism classes of principal G-bundles. This is a functor...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Basic Idea And Motivation
Chern classes are a characteristic class. They are topological invariants associated to vector bundles on a smooth manifold. If you descr...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - The Chern Class Of A Hermitian Vector Bundle On A Smooth Manifold
Given a complex hermitian vector bundle V of complex rank n over a smooth manifold M, a representative of each Chern class (also called a...   » Read the article

Encyclopedia - Chern Class: Encyclopedia Ii - Chern Class - Construction Of Chern Classes
There are various ways of approaching the subject. Each of which focuses on a slightly different flavor of Chern class. Originally Chern ...   » Read the article

Encyclopedia - Covariance And Contravariance: Encyclopedia Ii - Covariance And Contravariance - Algebra And Geometry
In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a ...   » Read the article