  The goal of local certification is to locally convince the vertices of a
graph $G$ that $G$ satisfies a given property. A prover assigns short
certificates to the vertices of the graph, then the vertices are allowed to
check their certificates and the certificates of their neighbors, and based
only on this local view, they must decide whether $G$ satisfies the given
property. If the graph indeed satisfies the property, all vertices must accept
the instance, and otherwise at least one vertex must reject the instance (for
any possible assignment of certificates). The goal is to minimize the size of
the certificates.
  In this paper we study the local certification of geometric and topological
graph classes. While it is known that in $n$-vertex graphs, planarity can be
certified locally with certificates of size $O(\log n)$, we show that several
closely related graph classes require certificates of size $\Omega(n)$. This
includes penny graphs, unit-distance graphs, (induced) subgraphs of the square
grid, 1-planar graphs, and unit-square graphs. These bounds are tight up to a
constant factor and give the first known examples of hereditary (and even
monotone) graph classes for which the certificates must have linear size. For
unit-disk graphs we obtain a lower bound of $\Omega(n^{1-\delta})$ for any
$\delta>0$ on the size of the certificates, and an upper bound of $O(n \log
n)$. The lower bounds are obtained by proving rigidity properties of the
considered graphs, which might be of independent interest.
