  The work of Reid, Chinburg--Hamilton--Long--Reid, Prasad--Rapinchuk, and the
author with Reid have demonstrated that geodesics or totally geodesic
submanifolds can sometimes be used to determine the commensurability class of
an arithmetic manifold. The main results of this article show that
generalizations of these results to other arithmetic manifolds will require a
wide range of data. Specifically, we prove that certain incommensurable
arithmetic manifolds arising from the semisimple Lie groups of the form
$(\SL(d,\R))^r \times (\SL(d,\C))^s$ have the same commensurability classes of
totally geodesic submanifolds coming from a fixed field. This construction is
algebraic and shows the failure of determining, in general, a central simple
algebra from subalgebras over a fixed field. This, in turn, can be viewed in
terms of forms of $\SL_d$ and the failure of determining the form via certain
classes of algebraic subgroups.
