The connection to the Anosov flow comes from the realization that formula_21 is the geodesic flow on "P" and "Q". Lie vector fields being (by definition) left invariant under the action of a group element, one has that these fields are left invariant under the specific elements formula_21 of the geodesic flow. In other words, the spaces "TP" and "TQ" are split into three one-dimensional spaces, or subbundles, each of which are invariant under the geodesic flow. The final step is to notice that vector fields in one subbundle expand (and expand exponentially), those in another are unchanged, and those in a third shrink (and do so exponentially).