Tangent bundle

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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 to each point of M

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

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

$\pi\colon TM \to M\,$

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

1 Topology and smooth structure
2 Examples
3 Vector fields
4 See also
5 External links
6 References

[edit] 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 vector space 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 T_xU to {x}× Rⁿ . 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. 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 (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 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

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

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

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

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

We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if $\tilde\phi_\alpha(A\cap U_\alpha)$ 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.

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.

[edit] 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.

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.

[edit] 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 the image of x, denoted V_x, lies in T_xM, the tangent space at 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

(f V) 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.

[edit] See also

[edit] External links

[edit] 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 3-540-42627-2
Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London. ISBN 0-8053-0102-X

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Categories: Differential topology | Vector bundles