The authors define the notion of geometric reductivity for arbitrary commutative Hopf algebras \(C\) as follows: for every comodule \(M\) and every comodule homomorphism \(\lambda \in M^*\) there is a symmetric power \(r\) such that \(S^ r(\lambda)\) does not vanish on \(S^ r(M)^ C\), the invariants of the \(r\)-th symmetric power of \(M\). The exponent of reductivity of a pair \((M,\lambda)\) is the minimal such \(r\). It is \(p^ n\) (in characteristic \(p\)) or 1. Properties of geometric reductivity under Hopf algebra maps are studied. Furthermore it is shown that every finitely generated commutative \(C\)-comodule algebra has a finitely generated algebra of invariants. This is applied to the dual of a restricted enveloping algebra of a (f.d.) restricted Lie algebra \(u^*({\mathcal L})\), which is geometrically reductive.