  Rollings of reductive homogeneous spaces are investigated. More precisely,
for a reductive homogeneous space $G / H$ with reductive decomposition
$\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$, we consider rollings of
$\mathfrak{m}$ over $G / H$ without slip and without twist, where $G / H$ is
equipped with an invariant covariant derivative. To this end, an intrinsic
point of view is taken, meaning that a rolling is a curve in the configuration
space $Q$ which is tangent to a certain distribution. By considering a
$H$-principal fiber bundle $\overline{\pi} \colon \overline{Q} \to Q$ over the
configuration space equipped with a suitable principal connection, rollings of
$\mathfrak{m}$ over $G / H$ can be expressed in terms of horizontally lifted
curves on $\overline{Q}$. The total space of $\overline{\pi} \colon
\overline{Q} \to Q$ is a product of Lie groups. In particular, for a given
control curve, this point of view allows for characterizing rollings of
$\mathfrak{m}$ over $G / H$ as solutions of an explicit, time-variant ordinary
differential equation (ODE) on $\overline{Q}$, the so-called kinematic
equation. An explicit solution for the associated initial value problem is
obtained for rollings with respect to the canonical invariant covariant
derivative of first and second kind if the development curve in $G / H$ is the
projection of a one-parameter subgroup in $G$. Lie groups and Stiefel manifolds
are discussed as examples.
