In representation theory, a branching rule describes the decomposition of the restriction of an irreducible representation to a subgroup. Let $i: F' \rightarrow F$ be the inclusion of a homogeneous variety in another homogeneous variety. The geometric analogue of the branching problem asks to calculate the induced map in cohomology in terms of the Schubert bases of $F$ and $F'$. In this talk, I will give a positive, geometric rule for computing the branching coefficients for the inclusion of an orthogonal flag variety in a Type-A flag variety. The geometric rule has many applications including to the restrictions of representations of $SL(n)$ to $SO(n)$, to the study of the moduli spaces of rank 2 vector bundles on hyperelliptic curves and to presentations of the cohomology ring of orthogonal flag varieties.