Vector Bundle
Introduction
A vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (called the "base" of the vector bundle). To be more specific, a vector bundle is a space E, together with a continuous surjective map π : E → X that, to each point x of the base X, associates a vector space (called the fiber over x) in such a way that the following conditions are met.
Definition
A vector bundle consists of the data (E, X, π, V), where E and X are topological spaces, π : E → X is a continuous surjection, and V is a real or complex vector space, such that for every x in X, there exists an open neighborhood U of x, and a homeomorphism φ : π−1(U) → U × V (where U × V is endowed with the product topology), such that the following diagram commutes:
Here, pr1 : U × V → U is the natural projection, and the map φ is called a (local) trivialization of the vector bundle. The dimension of the vector space V is called the rank of the vector bundle.
Examples
There are many examples of vector bundles in mathematics and physics. Here are a few:
1. The tangent bundle of a differentiable manifold is a vector bundle. The base space is the manifold itself, and the fiber over a point is the tangent space at that point.
2. The cotangent bundle of a differentiable manifold is also a vector bundle. The fiber over a point is the dual space to the tangent space at that point.
3. The normal bundle of a submanifold of a differentiable manifold is a vector bundle. The fiber over a point is the orthogonal complement of the tangent space at that point in the tangent space of the ambient manifold.
4. In physics, the space of all possible configurations of a system is often a vector bundle. The base space is the space of all possible positions of the system, and the fiber over a point is the space of all possible momenta at that position.
Properties
Vector bundles have many interesting properties. Here are a few:
1. Vector bundles are stable under several operations, such as taking direct sums, tensor products, and duals.
2. The set of isomorphism classes of rank n vector bundles over a topological space X forms an abelian group, called the n-th K-theory group of X.
3. The Chern classes of a complex vector bundle are topological invariants that can be used to distinguish non-isomorphic bundles.
4. The Atiyah-Singer index theorem relates the index of certain differential operators on a vector bundle to topological data of the bundle.