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In differential geometry, the tangent bundle of a
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the
disjoint union In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ...
The disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see
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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.
of the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
to M at the point x . So, an element of TM can be thought of as a pair (x,v), where x is a point in M and v is a tangent vector to M at x . There is a natural projection : \pi : TM \twoheadrightarrow M defined by \pi(x, v) = x. This projection maps each element of the tangent space T_xM to the single point x . The tangent bundle comes equipped with a natural topology (described in a section
below Below may refer to: *Earth * Ground (disambiguation) * Soil * Floor * Bottom (disambiguation) * Less than *Temperatures below freezing * Hell or underworld People with the surname * Ernst von Below (1863–1955), German World War I general * Fr ...
). With this topology, the tangent bundle to a manifold is the prototypical example of a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
(which is a
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
whose fibers are
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
s). A
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of TM is a vector field on M, and the dual bundle to TM is the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ...
, which is the disjoint union of the
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, ...
s of M . By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the
Whitney sum In mathematics, a vector bundle is a topology, topological construction that makes precise the idea of a family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, or an ...
TM\oplus E is trivial. For example, the ''n''-dimensional sphere ''Sn'' is framed for all ''n'', but parallelizable only for (by results of Bott-Milnor and Kervaire).


Role

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 f:M\rightarrow N is a smooth function, with M and N smooth manifolds, its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is a smooth function Df:TM\rightarrow TN .


Topology and smooth structure

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 TM is twice the dimension of M. Each tangent space of an ''n''-dimensional manifold is an ''n''-dimensional vector space. If U is an open
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
subset of M, then there is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
TU\to U\times\mathbb R^n which restricts to a linear isomorphism from each tangent space T_xU to \\times\mathbb R^n. As a manifold, however, TM is not always diffeomorphic to the product manifold M\times\mathbb R^n. When it is of the form M\times\mathbb R^n, 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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addit ...
. 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, tangent bundles are locally modeled on U\times\mathbb R^n, where U is an open subset of Euclidean space. If ''M'' is a smooth ''n''-dimensional manifold, then it comes equipped with an
atlas An atlas is a collection of maps; it is typically a bundle of maps of Earth or of a region of Earth. Atlases have traditionally been bound into book form, but today many atlases are in multimedia formats. In addition to presenting geograp ...
of charts (U_\alpha,\phi_\alpha), where U_\alpha is an open set in M and :\phi_\alpha: U_\alpha \to \mathbb R^n is a
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given tw ...
. These local coordinates on U_\alpha give rise to an isomorphism T_xM\rightarrow\mathbb R^n for all x\in U_\alpha. We may then define a map :\widetilde\phi_\alpha:\pi^\left(U_\alpha\right) \to \mathbb R^ by :\widetilde\phi_\alpha\left(x, v^i\partial_i\right) = \left(\phi_\alpha(x), v^1, \cdots, v^n\right) We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if :\widetilde\phi_\alpha\left(A\cap \pi^\left(U_\alpha\right)\right) is open in \mathbb R^ for each \alpha. These maps are homeomorphisms between open subsets of TM and \mathbb R^ and therefore serve as charts for the smooth structure on TM. The transition functions on chart overlaps \pi^\left(U_\alpha \cap U_\beta\right) are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of \mathbb R^. The tangent bundle is an example of a more general construction called a
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...
(which is itself a specific kind of
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a ...
). Explicitly, the tangent bundle to an n-dimensional manifold M may be defined as a rank n vector bundle over M whose transition functions are given by the
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant *Jacobian elliptic functions *Jacobian variety *Intermediate Jacobian In mathematics, the intermediate Jacobian of a compact Kähler m ...
of the associated coordinate transformations.


Examples

The simplest example is that of \mathbb R^n. In this case the tangent bundle is trivial: each T_x \mathbf \mathbb R^n is canonically isomorphic to T_0 \mathbb R^n via the map \mathbb R^n \to \mathbb R^n which subtracts x , giving a diffeomorphism T\mathbb R^n \to \mathbb R^n \times \mathbb R^n. Another simple example is the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
, S^1 (see picture above). The tangent bundle of the circle is also trivial and isomorphic to S^1\times\mathbb R . Geometrically, this is a
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infi ...
of infinite height. The only tangent bundles that can be readily visualized are those of the real line \mathbb R and the unit circle S^1, 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 S^2 : this tangent bundle is nontrivial as a consequence of the
hairy ball theorem The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, ...
. Therefore, the sphere is not parallelizable.


Vector fields

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 M is a smooth map :V\colon M \to TM such that V(x) = (x,V_x) with V_x\in T_xM for every x\in M. In the language of fiber bundles, such a map is called a ''
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''. A vector field on M is therefore a section of the tangent bundle of M. The set of all vector fields on M is denoted by \Gamma(TM). Vector fields can be added together pointwise :(V+W)_x = V_x + W_x and multiplied by smooth functions on ''M'' :(fV)_x = f(x)V_x to get other vector fields. The set of all vector fields \Gamma(TM) then takes on the structure of a module over the
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...
of smooth functions on ''M'', denoted C^(M). A local vector field on M is a ''local section'' of the tangent bundle. That is, a local vector field is defined only on some open set U\subset M and assigns to each point of U a vector in the associated tangent space. The set of local vector fields on M forms a structure known as a sheaf of real vector spaces on M. The above construction applies equally well to the cotangent bundle – the differential 1-forms on M are precisely the sections of the cotangent bundle \omega \in \Gamma(T^*M), \omega: M \to T^*M that associate to each point x \in M a 1-covector \omega_x \in T^*_xM, which map tangent vectors to real numbers: \omega_x : T_xM \to \R. Equivalently, a differential 1-form \omega \in \Gamma(T^*M) maps a smooth vector field X \in \Gamma(TM) to a smooth function \omega(X) \in C^(M).


Higher-order tangent bundles

Since the tangent bundle TM is itself a smooth manifold, the second-order tangent bundle can be defined via repeated application of the tangent bundle construction: :T^2 M = T(TM).\, In general, the kth order tangent bundle T^k M can be defined recursively as T\left(T^M\right). A smooth map f: M \rightarrow N has an induced derivative, for which the tangent bundle is the appropriate domain and range Df : TM \rightarrow TN. Similarly, higher-order tangent bundles provide the domain and range for higher-order derivatives D^k f : T^k M \to T^k N. A distinct but related construction are the jet bundles on a manifold, which are bundles consisting of jets.


Canonical vector field on tangent bundle

On every tangent bundle TM, considered as a manifold itself, one can define a canonical vector field V:TM\rightarrow T^2M 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, TW \cong W \times W, since the vector space itself is flat, and thus has a natural diagonal map W \to TW given by w \mapsto (w, w) 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 M is curved, each tangent space at a point x, T_x M \approx \mathbb^n, is flat, so the tangent bundle manifold TM is locally a product of a curved M and a flat \mathbb^n. Thus the tangent bundle of the tangent bundle is locally (using \approx for "choice of coordinates" and \cong for "natural identification"): :T(TM) \approx T(M \times \mathbb^n) \cong TM \times T(\mathbb^n) \cong TM \times ( \mathbb^n\times\mathbb^n) and the map TTM \to TM is the projection onto the first coordinates: :(TM \to M) \times (\mathbb^n \times \mathbb^n \to \mathbb^n). Splitting the first map via the zero section and the second map by the diagonal yields the canonical vector field. If (x,v) are local coordinates for TM, the vector field has the expression : V = \sum_i \left. v^i \frac \_. More concisely, (x, v) \mapsto (x, v, 0, v) – 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 v, not on x, as only the tangent directions can be naturally identified. Alternatively, consider the scalar multiplication function: :\begin \mathbb \times TM \to TM \\ (t,v) \longmapsto tv \end The derivative of this function with respect to the variable \mathbb R at time t=1 is a function V:TM\rightarrow T^2M , which is an alternative description of the canonical vector field. The existence of such a vector field on TM is analogous to the canonical one-form on the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ...
. Sometimes V is also called the Liouville vector field, or radial vector field. Using V one can characterize the tangent bundle. Essentially, V 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

There are various ways to
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobile ...
objects on M into objects on TM . For example, if \gamma is a curve in M , then \gamma' (the
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. Mo ...
of \gamma ) is a curve in TM . In contrast, without further assumptions on M (say, a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent spac ...
), there is no similar lift into the
cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ...
. The ''vertical lift'' of a function f:M\rightarrow\mathbb R is the function f^\vee:TM\rightarrow\mathbb R defined by f^\vee=f\circ \pi, where \pi:TM\rightarrow M is the canonical projection.


See also

* Pushforward (differential) * Unit tangent bundle *
Cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This ...
*
Frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts na ...
*
Musical isomorphism In mathematics—more specifically, in differential geometry—the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle \mathrmM and the cotangent bundle \mathrm^* M of a pseudo-Riemannian manifold induc ...


Notes


References

* . * John M. Lee, ''Introduction to Smooth Manifolds'', (2003) Springer-Verlag, New York. . * Jürgen Jost, ''Riemannian Geometry and Geometric Analysis'', (2002) Springer-Verlag, Berlin. * Ralph Abraham and Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London. * M. De León, E. Merino, J.A. Oubiña, M. Salgado, ''A characterization of tangent and stable tangent bundles'', Annales de l'institut Henri Poincaré (A) Physique théorique, Vol. 61, no. 1, 1994, 1-1


External links

*
Wolfram MathWorld: Tangent Bundle

PlanetMath: Tangent Bundle
{{Manifolds Differential topology Vector bundles