Tangent bundle


A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold is a manifold which assembles all the tangent vectors in . As a set, it is given by the disjoint union[note 1] of the tangent spaces of . That is,

Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).[note 1]

where denotes the tangent space to at the point . So, an element of can be thought of as a pair , where is a point in and is a tangent vector to at .

There is a natural projection

defined by . This projection maps each element of the tangent space to the single point .

The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (which is a fiber bundle whose fibers are vector spaces). A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of . By definition, a manifold is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold is framed if and only if the tangent bundle is stably trivial, meaning that for some trivial bundle the Whitney sum is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).

Role edit

One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if   is a smooth function, with   and   smooth manifolds, its derivative is a smooth function  .

Topology and smooth structure edit

The tangent bundle comes equipped with a natural topology (not the disjoint union topology) and smooth structure so as to make it into a manifold in its own right. The dimension of   is twice the dimension of  .

Each tangent space of an n-dimensional manifold is an n-dimensional vector space. If   is an open contractible subset of  , then there is a diffeomorphism   which restricts to a linear isomorphism from each tangent space   to  . As a manifold, however,   is not always diffeomorphic to the product manifold  . When it is of the form  , then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modeled on Euclidean space, tangent bundles are locally modeled on  , where   is an open subset of Euclidean space.

If M is a smooth n-dimensional manifold, then it comes equipped with an atlas of charts  , where   is an open set in   and


is a diffeomorphism. These local coordinates on   give rise to an isomorphism   for all  . We may then define a map




We use these maps to define the topology and smooth structure on  . A subset   of   is open if and only if


is open in   for each   These maps are homeomorphisms between open subsets of   and   and therefore serve as charts for the smooth structure on  . The transition functions on chart overlaps   are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of  .

The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). Explicitly, the tangent bundle to an  -dimensional manifold   may be defined as a rank   vector bundle over   whose transition functions are given by the Jacobian of the associated coordinate transformations.

Examples edit

The simplest example is that of  . In this case the tangent bundle is trivial: each   is canonically isomorphic to   via the map   which subtracts  , giving a diffeomorphism  .

Another simple example is the unit circle,   (see picture above). The tangent bundle of the circle is also trivial and isomorphic to  . Geometrically, this is a cylinder of infinite height.

The only tangent bundles that can be readily visualized are those of the real line   and the unit circle  , both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.

A simple example of a nontrivial tangent bundle is that of the unit sphere  : this tangent bundle is nontrivial as a consequence of the hairy ball theorem. Therefore, the sphere is not parallelizable.

Vector fields edit

A smooth assignment of a tangent vector to each point of a manifold is called a vector field. Specifically, a vector field on a manifold   is a smooth map


such that   with   for every  . In the language of fiber bundles, such a map is called a section. A vector field on   is therefore a section of the tangent bundle of  .

The set of all vector fields on   is denoted by  . Vector fields can be added together pointwise


and multiplied by smooth functions on M


to get other vector fields. The set of all vector fields   then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted  .

A local vector field on   is a local section of the tangent bundle. That is, a local vector field is defined only on some open set   and assigns to each point of   a vector in the associated tangent space. The set of local vector fields on   forms a structure known as a sheaf of real vector spaces on  .

The above construction applies equally well to the cotangent bundle – the differential 1-forms on   are precisely the sections of the cotangent bundle  ,   that associate to each point   a 1-covector  , which map tangent vectors to real numbers:  . Equivalently, a differential 1-form   maps a smooth vector field   to a smooth function  .

Higher-order tangent bundles edit

Since the tangent bundle   is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction:


In general, the  th order tangent bundle   can be defined recursively as  .

A smooth map   has an induced derivative, for which the tangent bundle is the appropriate domain and range  . Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives  .

A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.

Canonical vector field on tangent bundle edit

On every tangent bundle  , considered as a manifold itself, one can define a canonical vector field   as the diagonal map on the tangent space at each point. This is possible because the tangent space of a vector space W is naturally a product,   since the vector space itself is flat, and thus has a natural diagonal map   given by   under this product structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold   is curved, each tangent space at a point  ,  , is flat, so the tangent bundle manifold   is locally a product of a curved   and a flat   Thus the tangent bundle of the tangent bundle is locally (using   for "choice of coordinates" and   for "natural identification"):


and the map   is the projection onto the first coordinates:


Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field.

If   are local coordinates for  , the vector field has the expression


More concisely,   – the first pair of coordinates do not change because it is the section of a bundle and these are just the point in the base space: the last pair of coordinates are the section itself. This expression for the vector field depends only on  , not on  , as only the tangent directions can be naturally identified.

Alternatively, consider the scalar multiplication function:


The derivative of this function with respect to the variable   at time   is a function  , which is an alternative description of the canonical vector field.

The existence of such a vector field on   is analogous to the canonical one-form on the cotangent bundle. Sometimes   is also called the Liouville vector field, or radial vector field. Using   one can characterize the tangent bundle. Essentially,   can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.

Lifts edit

There are various ways to lift objects on   into objects on  . For example, if   is a curve in  , then   (the tangent of  ) is a curve in  . In contrast, without further assumptions on   (say, a Riemannian metric), there is no similar lift into the cotangent bundle.

The vertical lift of a function   is the function   defined by  , where   is the canonical projection.

See also edit

Notes edit

  1. ^ a b The disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S1, see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle.

References edit

  • Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, vol. 107, Providence: American Mathematical Society. ISBN 978-0-8218-4815-9
  • Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9981-8.
  • Jürgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3-540-42627-2
  • Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X
  • León, M. De; Merino, E.; Oubiña, J. A.; Salgado, M. (1994). "A characterization of tangent and stable tangent bundles" (PDF). Annales de l'I.H.P.: Physique Théorique. 61 (1): 1–15.
  • Gudmundsson, Sigmundur; Kappos, Elias (2002). "On the geometry of tangent bundles". Expositiones Mathematicae. 20: 1–41. doi:10.1016/S0723-0869(02)80027-5.

External links edit