  For a general spherically four--dimensional metric the notion of
"circularity" of a family of equatorial geodesic trajectories is defined in
geometrical terms. The main object turns out to be the angular--momentum
function $J$ obeying a consistency condition involving the mean extrinsic
curvature of the submanifold containing the geodesics. The ana\-ly\-sis of
linear stability is reduced to a simple dynamical system formally describing a
damped harmonic oscillator. For static metrics the existence of such geodesics
is given when $J^2 > 0$, and $(J^2)' > 0$ for stability. The formalism is then
applied to the Schwarzschild--de Sitter solution, both in its static and in its
time--dependent cosmological version, as well to the Kerr--de Sitter solution.
In addition we present an approximate solution to a cosmological metric
containing a massive source and solving the Einstein field equation for a
massless scalar.
