Tangent bundle

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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

<math>T(M) = \coprod_{x\in M}T_x(M).</math>

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

<math>\pi\colon T(M) \to M\,</math>

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

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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 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. Trival 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 (Uα, φα) where Uα is an open set in M and

<math>\phi_\alpha\colon U_\alpha \to \mathbb R^n</math>

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

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

by

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

We use these maps to define the topology and smooth structure on T(M). A subset A of T(M) is open iff <math>\tilde\phi_\alpha(A\cap U_\alpha)</math> 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 <math>\pi^{-1}(U_\alpha\cap U_\beta)</math> 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.

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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. 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.

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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

<math>V\colon M \to T(M)</math>

such that the image of x, denoted Vx, lies in Tx(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

<math>(V+W)_x = V_x + W_x</math>

and multiplied by smooth functions on M

<math>(fV)_x = f(x)V_x</math>

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.

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See also

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External links

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References

sv:Tangentknippe zh:切丛

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