# Tangent bundle

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*

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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

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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 it 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. As a set then, *T*(*M*) is isomorphic to *M* × **R**^{n}. As a manifold, however, *T*(*M*) is not always diffeomorphic to the product manifold *M* × **R**^{n}. When this happens the tangent bundle is said to be *trivial*. Just as manifolds are locally modelled on Euclidean space, tangent bundles are locally modelled on *M* × **R**^{n}.

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

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is a homeomorphism. These local coordinates on *U* give rise to an isomorphism between *T*_{x}*M* and **R**^{n} for each *x* ∈ *U*. We may then define a map

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by

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We use these maps to define the topology and smooth structure on *T*(*M*). A subset *A* of *T*(*M*) is open iff **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde\phi_\alpha(A\cap U_\alpha)}**
is open in **R**^{2n} for each α. These maps are then homeomorphisms between open subsets of *T*(*M*) and **R**^{2n} and therefore serve as charts for the smooth structure on *T*(*M*). The transition functions on chart overlaps **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 **R**^{2n}.

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

The simplest example is that of **R**^{n}. In this case the tangent bundle is trivial and isomorphic to **R**^{2n}. Another simple example is the unit circle, *S*^{1}. The tangent bundle is of the circle is also trivial and isomorphic to *S*^{1} × **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 *S*^{1}, 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 *S*^{2}. That the tangent bundle of *S*^{2} is nontrivial is a consequence of the hairy ball theorem.

## Vector fields

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

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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

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and multiplied by smooth functions on *M*

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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

## External links

## References

- 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 Jarrold E. Marsden,
*Foundations of Mechanics*, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X