In this paper we show that totally geodesic subspaces determine the
commensurability class of a standard arithmetic hyperbolic n-orbifold, nβ₯4. Many of the results are more general and apply to locally symmetric spaces
associated to arithmetic lattices in R-simple Lie groups of type
Bnβ and Dnβ. We use a combination of techniques from algebraic groups and
quadratic forms to prove several results about these spaces.Comment: 34 Pages. Corrected typos. Added references. Improved expositio