  We study the orbits and manifolds near the equilibrium points of a rotating
asteroid. The linearised equations of motion relative to the equilibrium points
in the gravitational field of a rotating asteroid, the characteristic equation
and the stable conditions of the equilibrium points are derived and discussed.
First, a new metric is presented to link the orbit and the geodesic of the
smooth manifold. Then, using the eigenvalues of the characteristic equation,
the equilibrium points are classified into 8 cases. A theorem is presented and
proved to describe the structure of the submanifold as well as the stable and
unstable behaviours of a massless particle near the equilibrium points. The
linearly stable, the non-resonant unstable, and the resonant equilibrium points
are discussed. There are three families of periodic orbits and four families of
quasi-periodic orbits near the linearly stable equilibrium point. For the
non-resonant unstable equilibrium points, there are four cases; for the
periodic orbit and the quasi-periodic orbit, the structures of the submanifold
and the subspace near the equilibrium points are studied for each case. For the
resonant equilibrium points, the dimension of the resonant manifold is greater
than four, and we find at least 1 family of periodic orbits near the resonant
equilibrium points. Besides, this theory is applied to asteroids 216 Kleopatra,
1620 Geographos, 4769 Castalia, and 6489 Golevka.
