  Pini and Vantini (2017) introduced the interval-wise testing procedure which
performs local inference for functional data defined on an interval domain,
where the output is an adjusted p-value function that controls for type I
errors. We extend this idea to a general setting where domain is a Riemannian
manifolds. This requires new methodology such as how to define adjustment sets
on product manifolds and how to approximate the test statistic when the domain
has non-zero curvature. We propose to use permutation tests for inference and
apply the procedure in three settings: a simulation on a "chameleon-shaped"
manifold and two applications related to climate change where the manifolds are
a complex subset of $S^2$ and $S^2 \times S^1$, respectively. We note the
tradeoff between type I and type II errors: increasing the adjustment set
reduces the type I error but also results in smaller areas of significance.
However, some areas still remain significant even at maximal adjustment.
