I am working with the Euclidean Group $SE(3)$ and a doubt came to my mind.
If we define a tangent bundle of $SE(3)$ is it this tangent bundle itself and Euclidean Space?
Thanks in advance for the help.
2026-05-14 11:33:00.1778758380
$SE(3)$ and its tangent bundle
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I am not sure to understand the question.
As a manifold, $SE(3) \cong SO(3) \times \Bbb R^3$. $SO(3) \cong \Bbb RP^3$, and is parallelizable so $TSE(3) \cong SE(3)\times \Bbb R^6$. But there is no natural structure of Euclidian space on $SE(3)$ since there is non natural addition here.