  We show that, for odd $d$, the $L^{\frac{d+2}2}$ bounds of Sogge and Xi for
the Nikodym maximal function over manifolds of constant sectional curvature,
are unstable with respect to metric perturbation, in the spirit of the work of
Sogge and Minicozzi. A direct consequence is the instability of the bounds for
the corresponding oscillatory integral operator. Furthermore, we extend our
construction to show that the same phenomenon appears for any $d$-dimensional
Riemannian manifold with a local totally geodesic submanifold of dimension
$\lceil{\frac{d+1}2}\rceil$ if $d\ge 3$. In contrast, Sogge's $L^\frac73$ bound
for the Nikodym maximal function on 3-dimensional variably curved manifolds is
stable with respect to metric perturbation.
