tangent bundle

Definition

The tangent bundle is a vector bundle that contains all the tangent spaces of all the points in the manifold. Or, if you like arbitrary unions,

Manifold structure

If is a smooth manifold with , then has a smooth manifold structure and has .

Tangent bundle on Lie groups

Let be a Lie group and be its Lie algebra. There is a (global) isomorphism of vector bundles (called the trivialization of the bundle)

where is the left-multiplication map by the element .

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References

@lee2013