Mathematical parametrization of vector spaces by another space
The (infinitely extended) Möbius strip is a line bundle over the 1-sphere S1. Locally around every point in S1, it looks like U × R (where U is an open arc including the point), but the total bundle is different from S1 × R (which is a cylinder instead).
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space (for example could be a topological space, a manifold, or an algebraic variety): to every point of the space we associate (or "attach") a vector space in such a way that these vector spaces fit together to form another space of the same kind as (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over .
The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space such that for all in : in this case there is a copy of for each in and these copies fit together to form the vector bundle over . Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial.
Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces.
In mathematics, a vectorbundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space...
tangent bundle of a differentiable manifold M {\displaystyle M} is a manifold T M {\displaystyle TM} which assembles all the tangent vectors in M {\displaystyle...
In mathematics, a holomorphic vectorbundle is a complex vectorbundle over a complex manifold X such that the total space E is a complex manifold and...
complex vectorbundle is a vectorbundle whose fibers are complex vector spaces. Any complex vectorbundle can be viewed as a real vectorbundle through...
vectorbundle is a (holomorphic or algebraic) vectorbundle that is stable in the sense of geometric invariant theory. Any holomorphic vectorbundle may...
information. Coherent sheaves can be seen as a generalization of vectorbundles. Unlike vectorbundles, they form an abelian category, and so they are closed under...
In mathematics, the tautological bundle is a vectorbundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle...
mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vectorbundle of all the cotangent spaces at every point in the manifold...
tangent bundle is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a vectorbundle of...
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vectorbundle E. The fiber of F(E) over a point x is the set of all...
setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are...
principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vectorbundle E {\displaystyle E} , which consists of all ordered bases of the vector space...
The pullback of a vectorbundle is a vectorbundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber...
theory of fiber bundles with a structure group G {\displaystyle G} (a topological group) allows an operation of creating an associated bundle, in which the...
g ) , {\displaystyle (M,g),\,} one defines the spinor bundle to be the complex vectorbundle π S : S → M {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf...
covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay...
the Chern classes are characteristic classes associated with complex vectorbundles. They have since become fundamental concepts in many branches of mathematics...
canonical vector-valued 1-form on the frame bundle, the torsion Θ {\displaystyle \Theta } of the connection form ω {\displaystyle \omega } is the vector-valued...
differentiable principal bundle or vectorbundle with a connection. Let G be a Lie group and P → M be a principal G-bundle on a smooth manifold M. Suppose...
covariant derivative. A connection form associates to each basis of a vectorbundle a matrix of differential forms. The connection form is not tensorial...
on any smooth fiber bundle. In particular, it does not rely on the possible vectorbundle structure of the underlying fiber bundle, but nevertheless, linear...
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