# Bundles

## Bundles Topics

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### Normal bundle

The normal bundle of a submanifold is the vector bundle over that consists of all pairs , where is in and is a vector in the vector quotient space . Provided has a Riemann metric, can be thought of as the orthogonal complement to .

### Complex vector bundle

A complex vector bundle is a vector bundle whose fiber bundle is a complex vector space. It is not necessarily a complex manifold, even if its base manifold is a complex manifold. If a complex vector bundle also has the structure of a complex manifold, and is holomorphic, then it is called a holomorphic vector bundle.

### Circle bundle

A circle bundle is a fiber bundle whose fibers are circles. It may also have the structure of a principal bundle if there is an action of that preserves the fibers, and is locally trivial. That is, if every point has a trivialization such that the action of on is the usual one.

### Vector bundle connection

A connection on a vector bundle is a way to "differentiate" bundle sections, in a way that is analogous to the exterior derivative of a function . In particular, a connection is a function from smooth sections to smooth sections of with one-forms that satisfies the following conditions. 1. (Leibniz rule), and 2. . Alternatively, a connection can be considered as a linear map from bundle sections of , i.e., a section of with a vector field , to sections of , in analogy to the directional derivative. The directional derivative of a function , in the direction of a vector field , is given by . The connection, along with a vector field , may be applied to a section of to get the section . From this perspective, connections must also satisfy(1)for any smooth function . This property follows from the first definition.For example, the trivial bundle admits a flat connection since any bundle section corresponds to a function . Then setting gives the connection...

### Vector bundle

A vector bundle is special class of fiber bundle in which the fiber is a vector space . Technically, a little more is required; namely, if is a bundle with fiber , to be a vector bundle, all of the fibers for need to have a coherent vector space structure. One way to say this is that the "trivializations" , are fiber-for-fiber vector space isomorphisms.A vector bundle is a total space along with a surjective map to a base manifold . Any fiber is a vector space isomorphic to .The simplest nontrivial vector bundle is a line bundleon the circle, and is analogous to the Möbius strip.One use for vector bundles is a generalization of vector functions. For instance, the tangent vectors of an -dimensional manifold are isomorphic to at a point in a coordinate chart. But the isomorphism with depends on the choice of coordinate chart. Nearby , the vector fields look like functions. To define vector fields on the whole manifold requires the tangent bundle,..

### Canonical bundle

The canonical bundle is a holomorphic line bundle on a complex manifold which is determined by its complex structure. On a coordinate chart , it is spanned by the nonvanishing section . The transition function between coordinate charts is given by the determinant of the Jacobian of the coordinate change.The canonical bundle is defined in a similar way to the holomorphic tangent bundle. In fact, it is the th exterior power of the dual bundle to the holomorphic tangent bundle.

### Holomorphic vector bundle

A complex vector bundle is a vector bundle whose fiber bundles are a copy of . is a holomorphic vector bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic. The simplest example is a holomorphic line bundle, where the fiber is simply a copy of .

### Bundle section

A section of a fiber bundle gives an element of the fiber over every point in . Usually it is described as a map such that is the identity on . A real-valued function on a manifold is a section of the trivial line bundle . Another common example is a vector field, which is a section of the tangent bundle.

### Holomorphic tangent bundle

The holomorphic tangent bundle to a complex manifold is given by its complexified tangent vectors which are of type . In a coordinate chart , the bundle is spanned by the local bundle sections . The antiholomorphic sections are spanned by , of type , where denotes the complex conjugate.

### Bundle rank

The rank of a vector bundle is the dimension of its fiber. Equivalently, it is the maximum number of linearly independent local bundle sections in a trivialization. Naturally, the dimension here is measured in the appropriate category. For instance, a real line bundle has fibers isomorphic with , and a complex line bundle has fibers isomorphic to , but in both cases their rank is 1.The rank of the tangent bundle of a real manifold is equal to the dimension of . The rank of a trivial bundle is equal to . There is no upper bound to the rank of a vector bundle over a fixed manifold .

### Holomorphic line bundle

A complex line bundle is a vector bundle whose fibers are a copy of . is a holomorphic line bundle if it is a holomorphic map between complex manifolds and its transition functions are holomorphic.On a compact Riemann surface, a variety divisor determines a line bundle. For example, consider on . Around there is a coordinate chart given by the holomorphic function with . Similarly, is a holomorphic function defining a disjoint chart around with . Then letting , the Riemann surface is covered by . The line bundle corresponding to is then defined by the following transition functions,(1)(2)

### Bundle orientation

A real vector bundle has an orientation if there exists a covering by trivializations such that the transition functions are vector space orientation-preserving. Alternatively, there exists a section of the projectivization of the top exterior power of the bundle, . A bundle is called orientable if there exists an orientation. Hence a bundle of bundle rank is orientable iff is a trivial line bundle.An orientation of the tangent bundle is equivalent to an orientation on the base manifold. Not all bundles are orientable, as can be seen by the tangent bundle of the Möbius strip. The nontrivial line bundle on the circle is also not orientable.

### Tangent map

If , then the tangent map associated to is a vector bundle homeomorphism (i.e., a map between the tangent bundles of and respectively). The tangent map corresponds to differentiation by the formula(1)where (i.e., is a curve passing through the base point to in at time 0 with velocity ). In this case, if and , then the chain rule is expressed as(2)In other words, with this way of formalizing differentiation, the chain rule can be remembered by saying that "the process of taking the tangent map of a map is functorial." To a topologist, the form(3)for all , is more intuitive than the usual form of the chain rule.

### Tangent bundle section

A vector field is a section of its tangent bundle, meaning that to every point in a manifold , a vector is associated, where is the tangent space.

### Bundle map

A bundle map is a map between bundles along with a compatible map between the base manifolds. Suppose and are two bundles, thenis a bundle map if there is a map such that for all . In particular, the fiber bundle of over a point , gets mapped to the fiber of over .In the language of category theory, the above diagram commutes. To be more precise, the induced map between fibers has to be a map in the category of the fiber. For instance, in a bundle map between vector bundles the fiber over is mapped to the fiber over by a linear transformation.For example, when is a smooth map between smooth manifolds then is the differential, which is a bundle map between the tangent bundles. Over any point in , the tangent vectors at get mapped to tangent vectors at by the Jacobian.

### Tangent bundle

Every smooth manifold has a tangent bundle , which consists of the tangent space at all points in . Since a tangent space is the set of all tangent vectors to at , the tangent bundle is the collection of all tangent vectors, along with the information of the point to which they are tangent.(1)The tangent bundle is a special case of a vector bundle. As a bundle it has bundle rank , where is the dimension of . A coordinate chart on provides a trivialization for . In the coordinates, ), the vector fields , where , span the tangent vectors at every point (in the coordinate chart). The transition function from these coordinates to another set of coordinates is given by the Jacobian of the coordinate change.For example, on the unit sphere, at the point there are two different coordinate charts defined on the same hemisphere, and ,(2)(3)with and . The map between the coordinate charts is .(4)The Jacobian of is given by the matrix-valued function(5)which has determinant..

### Frame bundle

The frame bundle on a Riemannian manifold is a principal bundle. Over every point , the Riemannian metric determines the set of orthonormal frames, i.e., the possible choices for an orthonormal basis for the tangent space . The collection of orthonormal frames is the frame bundle.The choice of an orthonormal frame at a point reflects a choice of coordinates, up to first order. Roughly speaking, the frame bundle reflects the ambiguity of choosing coordinates in Riemannian geometry. Consequently the frame bundle can be used to show that equations are well-defined, independent of coordinates, without any explicit reference to coordinates. A local bundle section of the frame bundle gives a moving frame, which can be used to calculate the classical tensors of differential geometry such as curvature.An orthogonal matrix acts on an orthonormal basis to give another orthonormal basis. Consequently, the frame bundle on a -dimensional manifold..

### Bundle

The term "bundle" is an abbreviated form of the full term fiber bundle. Depending on context, it may mean one of the special cases of fiber bundles, such as a vector bundle or a principal bundle. Bundles are so named because they contain a collection of objects which, like a bundle of hay, are held together in a special way. All of the fibers line up--or at least they line up to nearby fibers.Locally, a bundle looks like a product manifold in a trivialization. The graph of a function sits inside the product as . The bundle sections of a bundle generalize functions in this way. It is necessary to use bundles when the range of a function only makes sense locally, as in the case of a vector field on the sphere.Bundles are a special kind of sheaf.

### Finsler space

A general space based on the line elementwith for a function on the tangent bundle , and homogeneous of degree 1 in . Formally, a Finsler space is a smooth manifold possessing a Finsler metric. Finsler geometry is Riemannian geometry without the restriction that the line element be quadratic and of the formA compact boundaryless Finsler space is locally Minkowskian iffit has 0 "flag curvature."

### Finsler metric

A continuous real function defined on the tangent bundle of an -dimensional smooth manifold is said to be a Finsler metric if 1. is differentiable at , 2. for any element and any real number , 3. Denoting the metricthen is a positive definite matrix. A smooth manifold with a Finsler metric is called a Finsler space.

### Base manifold

The base manifold in a bundle is analogous to the domain for a set of functions. In fact, a bundle, by definition, comes with a map to the base manifold, often called or projection.For example, the base manifold to the tangent bundle of a manifold is the manifold . A vector field is a function from the manifold to the tangent bundle, with the restriction that every point gets mapped to a vector at that point. In general, a bundle has bundle sections, at least locally, which are maps from the base manifold to the bundle.

### Fibration

If is a fiber bundle with a paracompact topological space, then satisfies the homotopy lifting property with respect to all topological spaces. In other words, if is a homotopy from to , and if is a lift of the map with respect to , then has a lift to a map with respect to . Therefore, if you have a homotopy of a map into , and if the beginning of it has a lift, then that lift can be extended to a lift of the homotopy itself.A fibration is a map between topological spaces such that it satisfies the homotopy lifting property.

### Fiber bundle

A fiber bundle (also called simply a bundle) with fiber is a map where is called the total space of the fiber bundle and the base space of the fiber bundle. The main condition for the map to be a fiber bundle is that every point in the base space has a neighborhood such that is homeomorphic to in a special way. Namely, ifis the homeomorphism, thenwhere the map means projection onto the component. The homeomorphisms which "commute with projection" are called local trivializations for the fiber bundle . In other words, looks like the product (at least locally), except that the fibers for may be a bit "twisted."A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.Examples of fiber bundles include any product (which is a bundle over with fiber ), the Möbius strip (which..

### Associated vector bundle

Given a principal bundle , with fiber a Lie group and base manifold , and a group representation of , say , then the associated vector bundle is(1)In particular, it is the quotient space where .This construction has many uses. For instance, any group representation of the orthogonal group gives rise to a bundle of tensors on a Riemannian manifold as the vector bundle associated to the frame bundle.For example, is the frame bundle on , where(2)writing the special orthogonal matrix with rows . It is a bundle with the action defined by(3)which preserves the map .The tangent bundle is the associated vector bundle with the standard group representation of on , given by pairs , with and . Two pairs and represent the same tangent vector iff there is a such that and .

### Associated fiber bundle

Given a group action and a principal bundle , the associated fiber bundle on is(1)In particular, it is the quotient space where .For example, the torus has a action given by(2)and the frame bundle on the sphere,(3)is a principal bundle. The associated fiber bundle is a fiber bundle on the sphere, with fiber the torus. It is an example of a four-dimensional manifold.

### Dual bundle

Given a vector bundle , its dual bundle is a vector bundle . The fiber bundle of over a point is the dual vector space to the fiber of .

### Principal bundle

A principal bundle is a special case of a fiber bundle where the fiber is a group . More specifically, is usually a Lie group. A principal bundle is a total space along with a surjective map to a base manifold . Any fiber is a space isomorphic to . More specifically, acts freely without fixed point on the fibers, and this makes a fiber into a homogeneous space. For example, in the case of a circle bundle (i.e., when ), the fibers are circles, which can be rotated, although no point in particular corresponds to the identity. Near every point, the fibers can be given the group structure of in the fibers over a neighborhood by choosing an element in each fiber to be the identity element. However, the fibers cannot be given a group structure globally, except in the case of a trivial bundle.An important principal bundle is the frame bundle on a Riemannian manifold. This bundle reflects the different ways to give an orthonormal basis for tangent vectors.Consider all..

### Cotangent bundle

The cotangent bundle of a manifold is similar to the tangent bundle, except that it is the set where and is a dual vector in the tangent space to . The cotangent bundle is denoted .

### Anchor

An anchor is the bundle map from a vector bundle to the tangent bundle TB satisfying 1. and 2. where and are smooth sections of , is a smooth function of , and the bracket is the "Jacobi-Lie bracket" of a vector field.

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