  We compute the classical and quantum cohomology rings of the twistor spaces
of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum
multiplication by the first Chern class. Given a half-dimensional totally
geodesic submanifold we associate, after Reznikov, a monotone Lagrangian
submanifold of the twistor space. In the case of a 3-dimensional totally
geodesic submanifold of a hyperbolic 6-manifold we compute the obstruction term
$m_0$ in the Fukaya-Floer $A_{\infty}$-algebra of a Reznikov Lagrangian and
calculate the Lagrangian quantum homology. There is a well-known correspondence
between the possible values of $m_0$ for a Lagrangian with nonvanishing
Lagrangian quantum homology and eigenvalues for the action of $c_1$ on quantum
cohomology by quantum cup product. Reznikov's Lagrangians account for most of
these eigenvalues but there are four exotic eigenvalues we cannot account for.
