Editing Tangent bundle (section)

==Examples==
The simplest example is that of <math>\mathbb R^n</math>. In this case the tangent bundle is trivial: each <math> T_x \mathbf \mathbb R^n </math> is canonically isomorphic to <math> T_0 \mathbb R^n </math> via the map <math> \mathbb R^n \to \mathbb R^n </math> which subtracts <math> x </math>, giving a diffeomorphism <math> T\mathbb R^n \to \mathbb R^n \times \mathbb R^n</math>.

Another simple example is the [[unit circle]], <math> S^1 </math> (see picture above). The tangent bundle of the circle is also trivial and isomorphic to <math> S^1\times\mathbb R </math>. Geometrically, this is a [[cylinder (geometry)|cylinder]] of infinite height.

The only tangent bundles that can be readily visualized are those of the real line <math>\mathbb R </math> and the unit circle <math>S^1</math>, both of which are trivial. For 2-dimensional manifolds the tangent bundle is 4-dimensional and hence difficult to visualize.

A simple example of a nontrivial tangent bundle is that of the unit sphere <math> S^2 </math>: this tangent bundle is nontrivial as a consequence of the [[hairy ball theorem]]. Therefore, the sphere is not [[Parallelizable_manifold|parallelizable]].