Vector bundle

A vector bundle is a certain geometric construction corresponding to a family of vector spaces parametrized by another space $X$ ${\ displaystyle X}$ $X$ (eg, $X$ ${\ displaystyle X}$ $X$ can be a topological space , a variety, or an algebraic structure ): each point $x$ ${\ displaystyle x}$ $x$ of space $X$ ${\ displaystyle X}$ $X$ vector space is mapped $V_{x}$ ${\ displaystyle V_ {x}}$ $V_ {x}$ so that their union forms a space of the same type as $X$ ${\ displaystyle X}$ $X$ (topological space, variety or algebraic structure, etc.), called the space of a vector bundle over $X$ ${\ displaystyle X}$ $X$ . Space itself $X$ ${\ displaystyle X}$ $X$ called the base of the bundle .

The vector bundle is a special type of locally trivial bundles , which in turn are a special type of bundles .

Usually consider vector spaces over real or complex numbers. In this case, vector bundles are called real or complex, respectively. Complex vector bundles can be considered as real with an additionally introduced structure.

Content

Examples

The simplest example is the trivial bundle , which has the form of a direct product $X\times V$ ${\ displaystyle X \ times V}$ where $X$ ${\ displaystyle X}$ Is the topological space (the base of the bundle), and $V$ ${\ displaystyle V}$ vector space.
A more complex example is the tangent bundle of a smooth manifold : each point on the manifold is associated with a tangent space to the manifold at this point. The tangent bundle in the general case can be nontrivial.

Definitions

A vector bundle is a locally trivial bundle with a fiber $V$ ${\ displaystyle V}$ is a vector space with a structural group of reversible linear transformations $V$ ${\ displaystyle V}$ .

Related Definitions

A subbundle U of a vector bundle V on a topological space X is such a collection of linear subspaces $U_{x}\subset V_{x}$ ${\ displaystyle U_ {x} \ subset V_ {x}}$ , $x\in X$ ${\ displaystyle x \ in X}$ , which itself has the structure of a vector bundle.
A line bundle is a rank 1 vector bundle.

Morphisms

Morphism from a vector bundle $\pi _{1}\colon E_{1}\to X_{1}$ ${\ displaystyle \ pi _ {1} \ colon E_ {1} \ to X_ {1}}$ into vector bundle $\pi _{2}\colon E_{2}\to X_{2}$ ${\ displaystyle \ pi _ {2} \ colon E_ {2} \ to X_ {2}}$ defined by a pair of continuous mappings $f\colon E_{1}\to E_{2}$ ${\ displaystyle f \ colon E_ {1} \ to E_ {2}}$ and $g\colon X_{1}\to X_{2}$ ${\ displaystyle g \ colon X_ {1} \ to X_ {2}}$ such that

$g\circ \pi _{1}=\pi _{2}\circ f$ ${\ displaystyle g \ circ \ pi _ {1} = \ pi _ {2} \ circ f}$
for anyone $x\in X_{1}$ ${\ displaystyle x \ in X_ {1}}$ mapping $\pi _{1}^{-1}(\{x\})\to \pi _{2}^{-1}(\{g(x)\}),$ ${\ displaystyle \ pi _ {1} ^ {- 1} (\ {x \}) \ to \ pi _ {2} ^ {- 1} (\ {g (x) \}),}$ induced $f,$ ${\ displaystyle f,}$ - linear mapping of vector spaces.

notice, that $g$ ${\ displaystyle g}$ determined by $f$ ${\ displaystyle f}$ (because $\pi _{1}$ ${\ displaystyle \ pi _ {1}}$ - surjection), in which case they say that $f$ ${\ displaystyle f}$ covers $g$ ${\ displaystyle g}$ .

The class of all vector bundles together with morphisms of bundles forms a category . Restricting ourselves to vector bundles, which are smooth manifolds, and smooth bundle morphisms, we obtain the category of smooth vector bundles . Morphisms of vector bundles are a special case of the mapping of bundles between locally trivial bundles; they are often called a homomorphism of (vector) bundles .

A homomorphism of bundles from $E_{1}$ ${\ displaystyle E_ {1}}$ at $E_{2}$ ${\ displaystyle E_ {2}}$ , together with the inverse homomorphism, is called an isomorphism of (vector) bundles . In this case, the bundles $E_{1}$ ${\ displaystyle E_ {1}}$ and $E_{2}$ ${\ displaystyle E_ {2}}$ called isomorphic . Isomorphism of a vector bundle (rank $k$ ${\ displaystyle k}$ ) $E$ ${\ displaystyle E}$ above $X$ ${\ displaystyle X}$ on the trivial bundle (rank $k$ ${\ displaystyle k}$ above $X$ ${\ displaystyle X}$ ) is called trivialization $E$ ${\ displaystyle E}$ , wherein $E$ ${\ displaystyle E}$ called trivial (or trivializable ). It follows from the definition of a vector bundle that any vector bundle is locally trivial .

Bundle Operations

Most operations on vector spaces can be continued to vector bundles, carried out pointwise .

For example, if $E$ ${\ displaystyle E}$ - vector bundle into $X$ ${\ displaystyle X}$ then there is a bundle $E^{*}$ ${\ displaystyle E ^ {*}}$ on $X$ ${\ displaystyle X}$ called conjugate bundle whose fiber at the point $x\in X$ ${\ displaystyle x \ in X}$ Is the conjugate vector space $(E_{x})^{*}$ ${\ displaystyle (E_ {x}) ^ {*}}$ . Formally $E^{*}$ ${\ displaystyle E ^ {*}}$ can be defined as many pairs $(x,\varphi )$ ${\ displaystyle (x, \ varphi)}$ where $x\in X$ ${\ displaystyle x \ in X}$ and $\varphi \in E_{x}^{*}$ ${\ displaystyle \ varphi \ in E_ {x} ^ {*}}$ . The conjugate bundle is locally trivial.

There are many functorial operations performed on pairs of vector spaces (on one field). They can be directly extended to pairs of vector bundles $E, F$ ${\ displaystyle E, F}$ on $X$ ${\ displaystyle X}$ (above the given field). Here are some examples.

Whitney sum , or direct sum bundle $E$ ${\ displaystyle E}$ and $F$ ${\ displaystyle F}$ Is a vector bundle $E\oplus F$ ${\ displaystyle E \ oplus F}$ on $X$ ${\ displaystyle X}$ whose layer is at a point $x$ ${\ displaystyle x}$ is a direct sum $E_{x}\oplus F_{x}$ ${\ displaystyle E_ {x} \ oplus F_ {x}}$ vector spaces $E_{x}$ ${\ displaystyle E_ {x}}$ and $F_{x}$ ${\ displaystyle F_ {x}}$ .

Bundle of tensor product $E\otimes F$ ${\ displaystyle E \ otimes F}$ is defined similarly using pointwise tensor products of vector spaces.

Stratification of homomorphisms ( hom-bundle ) $\operatorname {Hom} \,(E,F)$ ${\ displaystyle \ operatorname {Hom} \, (E, F)}$ Is a vector bundle whose layer at a point $x$ ${\ displaystyle x}$ Is the space of linear mappings from $E_{x}$ ${\ displaystyle E_ {x}}$ at $F_{x}$ ${\ displaystyle F_ {x}}$ (often referred to $\operatorname {Hom} \,(E_{x},F_{x})$ ${\ displaystyle \ operatorname {Hom} \, (E_ {x}, F_ {x})}$ or $L(E_{x},F_{x})$ ${\ displaystyle L (E_ {x}, F_ {x})}$ ) This bundle is useful because there exists a bijection between the homomorphisms of vector bundles from $E$ ${\ displaystyle E}$ at $F$ ${\ displaystyle F}$ on $X$ ${\ displaystyle X}$ and parts $\operatorname {Hom} \,(E,F)$ ${\ displaystyle \ operatorname {Hom} \, (E, F)}$ on $X$ ${\ displaystyle X}$ .

Links

Mishchenko A.S. Vector bundles and their applications. - M .: Science. Chap. ed. Phys.-Math. lit., 1984. - 208 p.
Jurgen Jost. Riemannian Geometry and Geometric Analysis - (2002) Springer-Verlag, Berlinб ISBN 3-540-42627-2 - See section 1.5 .
. Jerrold E. Marsden. Foundations of Mechanics, - (1978) Benjamin-Cummings, Londonб ISBN 0-8053-0102-X - See section 1.5 .