Lie groups as spaces with trivial tangent bundle

Question

I have read in Frankel's differential geometry book the statement that spaces with trivial tangent bundles are Lie groups, which was given without proof.

How can I see that this is true intuitively and rigorously?

Does this assertion work in both directions?

Answer 1

A Lie group $G$ necessarily has a trivial tangent bundle because left multiplication $L_g$ by an element $g\in G$ induces a map identifying the tangent space at the identity with the tangent space at $g$, and these identifications are continuous in $g$. The converse is not true.