Examples
The simplest example is that of Rn. In this case the tangent bundle is trivial.
Another simple example is the unit circle, S1 (see picture above). The tangent bundle of the circle is also trivial and isomorphic to S1 × R. Geometrically, this is a cylinder of infinite height (see the bottom picture).
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 difficult to visualize.
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. Therefore, the sphere is not parallelizable.
Read more about this topic: Tangent Bundle
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—Bernard Mandeville (1670–1733)
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