Tangent Bundle - Topology and Smooth Structure

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 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 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 for manifolds equipped with a 'compatible group structure'; for instance, in the case where the manifold is a Lie group. The tangent bundle of the unit circle is trivial because it is a Lie group (under multiplication and its natural differential structure). It is not true however that all spaces with trivial tangent bundles are Lie groups; manifolds which have a trivial tangent bundle are called parallelizable. Just as manifolds are locally modelled on 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 (U_α, φ_α) where U_α is an open set in M and

is a diffeomorphism. 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

by

We use these maps to define the topology and smooth structure on TM. A subset A of TM is open if and only if 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 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.