To compare the lengths of vectors in formula_29 at different points "q", one needs a metric. Any inner product at formula_30 extends to a left-invariant Riemannian metric on "P", and thus to a Riemannian metric on "Q". The length of a vector formula_31 expands exponentially as exp(t) under the action of formula_21. The length of a vector formula_33 shrinks exponentially as exp(-t) under the action of formula_21. Vectors in formula_35 are unchanged. This may be seen by examining how the group elements commute. The geodesic flow is invariant,