In this paper, we consider two cases of rolling of one smooth connected
complete Riemannian manifold (M,g) onto another one (\hM,\hg) of equal
dimension n≥2. The rolling problem (NS) corresponds to the situation
where there is no relative spin (or twist) of one manifold with respect to the
other one. As for the rolling problem (R), there is no relative spin and also
no relative slip. Since the manifolds are not assumed to be embedded into an
Euclidean space, we provide an intrinsic description of the two constraints
"without spinning" and "without slipping" in terms of the Levi-Civita
connections ∇g and \nabla^{\hg}. For that purpose, we recast the
two rolling problems within the framework of geometric control and associate to
each of them a distribution and a control system. We then investigate the
relationships between the two control systems and we address for both of them
the issue of complete controllability. For the rolling (NS), the reachable
set (from any point) can be described exactly in terms of the holonomy groups
of (M,g) and (\hM,\hg) respectively, and thus we achieve a complete
understanding of the controllability properties of the corresponding control
system. As for the rolling (R), the problem turns out to be more delicate. We
first provide basic global properties for the reachable set and investigate the
associated Lie bracket structure. In particular, we point out the role played
by a curvature tensor defined on the state space, that we call the
\emph{rolling curvature}. In the case where one of the manifolds is a space
form (let say (\hM,\hg)), we show that it is enough to roll along loops of
(M,g) and the resulting orbits carry a structure of principal bundle which
preserves the rolling (R) distribution. In the zero curvature case, we deduce
that the rolling (R) is completely controllable if and only if the holonomy
group of (M,g) is equal to SO(n). In the nonzero curvature case, we prove
that the structure group of the principal bundle can be realized as the
holonomy group of a connection on TM⊕R, that we call the rolling
connection. We also show, in the case of positive (constant) curvature, that if
the rolling connection is reducible, then (M,g) admits, as Riemannian
covering, the unit sphere with the metric induced from the Euclidean metric of
Rn+1. When the two manifolds are three-dimensional, we provide a complete
local characterization of the reachable sets when the two manifolds are
three-dimensional and, in particular, we identify necessary and sufficient
conditions for the existence of a non open orbit. Besides the trivial case
where the manifolds (M,g) and (\hM,\hg) are (locally) isometric, we show
that (local) non controllability occurs if and only if (M,g) and (\hM,\hg)
are either warped products or contact manifolds with additional restrictions
that we precisely describe. Finally, we extend the two types of rolling to the
case where the manifolds have different dimensions