A homogeneous fibration F of a riemannian manifold M is a fibration on which the group of isometries of M acts transitively. In other words, for any two fibers, some isometry of M takes one to the other while taking fibers to fibers. By contrast, a parallel fibration is one where all the fibers are a constant distance apart: the distance from one fiber to a point on a second fiber does not depend on the choice of the point in the second fiber.

I will talk about the relationship between these two properties: whether, in various contexts (i.e. imposing various conditions on the manifold M or the fibers) parallel fibrations are necessarily homogeneous or vice versa.