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

$T(M)\; =\; \backslash coprod\_\{x\backslash in\; M\}T\_x(M).$

An element of T(M) is a pair (x,v) where x ∈ M and v ∈ T_x(M), the tangent space at x. There is a natural projection

$\backslash pi\backslash colon\; T(M)\; \backslash to\; M\backslash ,$

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

Topology and smooth structure

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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 T(M) is twice the dimension of M.

Each tangent space of an n-dimensional vector space is an n-dimensional vector space. If U is an open contractable subset of M, then T(U) is homeomorphic to U × Rⁿ such that the map on the vector space structure is an isomorphism and diffeomorphic if the differentiable structure is defined. As a manifold, however, T(M) is not always diffeomorphic to the product manifold M × Rⁿ. When it is of the form M × Rⁿ, then we say that the tangent bundle is a trivial bundle. Trivial tangent bundles usually occur with trivial topological spaces or when there is special group action. 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. Just as manifolds are locally modelled on Euclidean space, tangent bundles are locally modelled on U × Rⁿ, where U is an open contractable subset in the manifold.

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

$\backslash phi\_\backslash alpha\backslash colon\; U\_\backslash alpha\; \backslash to\; \backslash mathbb\; R^n$

is a homeomorphism. These local coordinates on U give rise to an isomorphism between T_xM and Rⁿ for each x ∈ U. We may then define a map

$\backslash tilde\backslash phi\_\backslash alpha\backslash colon\; \backslash pi^\{-1\}(U\_\backslash alpha)\; \backslash to\; \backslash mathbb\; R^\{2n\}$

by

$\backslash tilde\backslash phi\_\backslash alpha(x,\; v^i\backslash partial\_i)\; =\; (\backslash phi\_\backslash alpha(x),\; v^1,\; \backslash cdots,\; v^n)$

We use these maps to define the topology and smooth structure on T(M). A subset A of T(M) is open if and only if $\backslash tilde\backslash phi\_\backslash alpha(A\backslash cap\; U\_\backslash alpha)$ is open in R²ⁿ for each α. These maps are then homeomorphisms between open subsets of T(M) and R²ⁿ and therefore serve as charts for the smooth structure on T(M). The transition functions on chart overlaps $\backslash pi^\{-1\}(U\_\backslash alpha\backslash cap\; U\_\backslash beta)$ are induced by the Jacobian matrices of the associated coordinate transformation and are therefore smooth maps between open subsets of R²ⁿ.

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 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.

Examples

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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. The tangent bundle of the circle is also trivial and isomorphic to S¹ × R. Geometrically, this is a cylinder of infinite height.

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.

Perhaps the simplest example of a nontrivial tangent bundle is that of the unit sphere S²: that this tangent bundle is nontrivial is a consequence of the hairy ball theorem.

Vector fields

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

$V\backslash colon\; M\; \backslash to\; T(M)$

such that the image of x, denoted V_x, lies in T_x(M), the tangent space to x. In the language of fiber bundles, such a map is called a section. 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

$(V+W)\_x\; =\; V\_x\; +\; W\_x$

and multiplied by smooth functions on M

$(fV)\_x\; =\; f(x)V\_x$

to get other vector fields. The set of all vector fields Γ(TM) then takes on the structure of a module over the commutative algebra 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 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.

See also

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• pushforward
• vector field
• vector bundle
• cotangent bundle
• frame bundle

External links

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• MathWorld: Tangent Bundle
• PlanetMath: Tangent Bundle

References

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• John M. Lee, Introduction to Smooth Manifolds, (2003) Springer-Verlag, New York. ISBN 0-387-95495-3.
• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin. ISBN 3540426272
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X

Differential topology | Fiber bundles

Fibrado tangente | Tangentknippe | 切丛