  We prove a radial maximal function characterisation of the local atomic Hardy
space h^1(M) on a Riemannian manifold M with positive injectivity radius and
Ricci curvature bounded from below. As a consequence, we show that an
integrable function belongs to h^1(M) if and only if either its local heat
maximal function or its local Poisson maximal function are integrable. A key
ingredient is a decomposition of H\"older cut-offs in terms of an appropriate
class of approximations of the identity, which we obtain on arbitrary
Ahlfors-regular metric measure spaces and generalises a previous result of A.
Uchiyama.
