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

In mathematics, the tangent bundle of a smooth (or differentiable) manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces of the points of M

TM = \coprod_{x\in M}T_xM.

An element of TM is a pair (x,v) where x ∈ M and v ∈ TxM, the tangent space at x. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since There is a natural projection

\pi\colon TM \to M\,

which sends (x,v) to the base point x.

Contents

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. In General topology and related areas of Mathematics, the disjoint union (also called the direct sum, free union, or coproduct) of In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential 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 subset of M, then there is a diffeomorphism from TU to U × Rn which restricts to a linear isomorphism from each tangent space TxU to {x}× Rn . In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable As a manifold, however, TM is not always diffeomorphic to the product manifold M × Rn. When it is of the form M × Rn, then the tangent bundle is said to be trivial. Trivial tangent bundles usually occur with trivial topological spaces or when there is special group action. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. For instance, in the case where the manifold is an open set in Rn. Also the tangent bundle of the unit circle is trivial because it is a Lie group which acts on itself. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Just as manifolds are locally modelled on (open subsets of) Euclidean space, tangent bundles are locally modelled on U × Rn, where U is an open subset of Euclidean space.

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

\phi_\alpha\colon U_\alpha \to \mathbb R^n

is a diffeomorphism. For other uses of "atlas" see Atlas (disambiguation. In Mathematics, particularly topology an atlas describes how In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable These local coordinates on U give rise to an isomorphism between TxM and Rn for each x ∈ U. We may then define a map

\tilde\phi_\alpha\colon \pi^{-1}(U_\alpha) \to \mathbb R^{2n}

by

\tilde\phi_\alpha(x, v^i\partial_i) = (\phi_\alpha(x), v^1, \cdots, v^n)

We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if \tilde\phi_\alpha(A\cap \pi^{-1}(U_\alpha)) is open in R2n for each α. These maps are then homeomorphisms between open subsets of TM and R2n and therefore serve as charts for the smooth structure on TM. The transition functions on chart overlaps \pi^{-1}(U_\alpha\cap U_\beta) are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of R2n. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

The tangent bundle is an example of a more general construction called a vector bundle (which is itself a specific kind of fiber bundle). In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space In Mathematics, in particular in Topology, a fiber bundle (or fibre bundle) is a space which looks locally like a Product space. 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 of the associated coordinate transformations. In Vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its Determinant, the Jacobian determinant.

Examples

The simplest example is that of Rn. In this case the tangent bundle is trivial and isomorphic to R2n. Another simple example is the unit circle, S1. In Mathematics, a unit circle is The tangent bundle of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder of infinite height (see the picture on top right). A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis

Unfortunately, the only tangent bundles that can be readily visualized are those of the real line R and the unit circle S1, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence not easily visualizable.

A simple example of a nontrivial tangent bundle is that of the unit sphere S2: this tangent bundle is nontrivial as a consequence of the hairy ball theorem. The hairy ball theorem of Algebraic topology states that there is no nonvanishing continuous Tangent vector field on the sphere

Vector fields

A smooth assignment of a tangent vector to each point of a manifold is called a vector field. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. Specifically, a vector field on a manifold M is a smooth map

V\colon M \to TM

such that the image of x, denoted Vx, lies in TxM, the tangent space at x. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In the language of fiber bundles, such a map is called a section. In the Mathematical field of Topology, a section (or cross section) of a Fiber bundle, &pi: E &rarr B 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 Γ(TM). Vector fields can be added together pointwise

(V + W)x = Vx + Wx

and multiplied by smooth functions on M

(fV)x = f(x)Vx

to get other vector fields. The set of all vector fields Γ(TM) then takes on the structure of a module over the commutative algebra of smooth functions on M, denoted C∞(M). In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings

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 in 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. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space.

Canonical vector field on TM

On every tangent bundle TM one can define a canonical vector field. If (x, v) are local coordinates for TM, the vector field has the expression

V = \sum_i \left. v^i \frac{\partial}{\partial v^i} \right|_{(x,v)}.

Let us point out that V acts as a map V\colon TM\to TTM. One can also show that V does not depend on the local coordinates chosen for TM.

The existence of such a vector field on TM can be compared with the existence of a canonical 1-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 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 objects on M into objects on TM. For example, if c is a curve in M, then c' (the tangent of c) is a curve in TM. For the tangent function see Trigonometric functions. For other uses see Tangent (disambiguation. Let us point out that without further assumptions on M (say, a Riemannian metric), there is no similar lift into the cotangent bundle. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces

The vertical lift of a function f\colon M\to \mathbb{R} is the function f^v\colon TM\to \mathbb{R} defined by f^v=f\circ \pi, where \pi\colon TM\to M is the canonical projection.

Higher-Order Tangent Bundles

A second-order tangent bundle can be defined via repeated application of the tangent bundle:

T2M = T(TM).

Just as transition functions on chart overlaps are induced by the Jacobian matrices of the corresponding transition functions, second order maps are induced by the tensor

J_{i,j,k} = \frac{\partial \phi_\alpha^i}{\partial u^j \partial u^k}.

In general, the kth-order tangent bundle TkM can be defined inductively as T(Tk − 1M). Informally these higher-order tangents can be thought of as higher-order derivatives.

See also

References

External links

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