One starts by noting that T1H is isomorphic to the Lie group which have the algebra The exponential maps define right-invariant flows on the manifold of T1H=PSL(2,R), and likewise on T1M. Defining P=T1H and Q=T1M, these flows define vector fields on P and Q, whose vectors lie in TP and TQ. These are just the standard, ordinary Lie vector fields on the manifold of a Lie group, and the presentation above is a standard exposition of a Lie vector field.
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