HOME

TheInfoList



Click Here for Items Related To - flat vector bundle
OR:
flat vector bundle on:   [Wikipedia]   [Google]   [Amazon]

In mathematics, a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
is said to be ''flat'' if it is endowed with a linear connection with vanishing curvature, i.e. a
flat connection In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie alge ...
.


de Rham cohomology of a flat vector bundle

Let \pi:E \to X denote a flat vector bundle, and \nabla : \Gamma(X, E) \to \Gamma\left(X, \Omega_X^1 \otimes E\right) be the covariant derivative associated to the flat connection on E. Let \Omega_X^* (E) = \Omega^*_X \otimes E denote the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
(in fact a
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper se ...
of
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
s over \mathcal O_X) of differential forms on ''X'' with values in ''E''. The covariant derivative defines a degree-1
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
''d'', the differential of \Omega_X^*(E), and the flatness condition is equivalent to the property d^2 = 0. In other words, the graded vector space \Omega_X^* (E) is a
cochain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of th ...
. Its cohomology is called the de Rham cohomology of ''E'', or de Rham cohomology with coefficients twisted by the local coefficient system ''E''.


Flat trivializations

A trivialization of a flat vector bundle is said to be flat if the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
vanishes in this trivialization. An equivalent definition of a flat bundle is the choice of a trivializing atlas with locally constant transition maps.


Examples

* Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over \mathbb C\backslash \, with the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms. Historically, connection forms were introduced by Élie Carta ...
s 0 and -\frac\frac. The parallel vector fields are constant in the first case, and proportional to local determinations of the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . ...
in the second. * The real
canonical line bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it ...
\Lambda^{\mathrm{topM of a
differential manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
''M'' is a flat line bundle, called the orientation bundle. Its sections are
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...
s. * A Riemannian manifold is flat if and only if its Levi-Civita connection gives its tangent vector bundle a flat structure.


See also

*
Vector-valued differential forms 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 diff ...
*
Local system In mathematics, a local system (or a system of local coefficients) on a topological space ''X'' is a tool from algebraic topology which interpolates between cohomology with coefficients in a fixed abelian group ''A'', and general sheaf cohomology ...
, the more general notion of a locally constant sheaf. * Orientation character, a characteristic form related to the orientation line bundle, useful to formulate
Twisted Poincaré duality In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local information, by means of a local coefficient ...
*
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
whose connected component, the
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
, is the moduli space of algebraic flat line bundles. * Monodromy, or
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the fundamental group by
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
on flat bundles. *
Holonomy In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
, the obstruction to flatness. Vector bundles