We discuss the problem of testing hypotheses on functional data defined on manifold domain. Aim of this work is to test hypotheses locally, i.e., to provide a p-value function \( \tilde{p} \) (s) that can be used to test a null hypothesis against an alternative one on every point of the manifold. In the following we present three methods to do so: an adjusted p-value function controlling the functional false discovery rate; an adjusted p-value function controlling the ball-wise error rate; an adjusted e-value function that is ball-wise e-valid.