  Consider a smooth manifold and an action on it of a compact connected Lie
group with a bi-invariant metric. Then, any orbit is an embedded submanifold
that is isometric to a normal homogeneous space for the group. In this paper,
we establish new explicit and intrinsic formulas for the geometry of any such
orbit. We derive our formula of the Levi-Civita connection from an existing
generic formula for normal homogeneous spaces, i.e. which determines, a priori
only theoretically, the connection. We say that our formulas are effective:
they are directly usable, notably in numerical analysis, provided that the
ambient manifold is convenient for computations. Then, we deduce new effective
formulas for flag manifolds, since we prove that they are orbits under a
suitable action of the special orthogonal group on a product of Grassmannians.
This representation of them is quite useful, notably for studying flags of
eigenspaces of symmetric matrices. Thus, we develop from it a geometric
solution to the problem of perturbation of eigenvectors which is explicit and
optimal in a certain sense, improving the classical analytic solution.
