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Is there an obvious example of a nonvanishing vector field on SLn(R)? It seems like there should be a simple (constant?) one since it is a Lie group.

Thanks :)

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You can take any zero trace matrix $X$ and define (the left invariant extension) vector field $\tilde{X}$ on $\mathrm{SL}_n(\mathbb{R})$ via $\forall g\in\mathrm{SL}_n(\mathbb{R}),\tilde{X}_g:=gX$ If you take a basis $X_1,\dots,X_{n^2-1}$ of $\mathrm{T}_{id}\mathrm{SL}_n(\mathbb{R})\approx \mathrm{Ker}(Tr),$ this procedure yields a family of vector fields on $\mathrm{SL}_n(\mathbb{R})$ that are everywhere nonvanishing and actually form a basis of the tangent spaces at all all points $g\in\mathrm{SL}_n(\mathbb{R})$. This is generally true for all Lie groups. This is how you show they are parallelizable.