  In this work, we pose and solve the time-optimal navigation problem
considered on a slippery mountain slope modeled by a Riemannian manifold of an
arbitrary dimension, under the action of a cross gravitational wind. The impact
of both lateral and longitudinal components of gravitational wind on the time
geodesics is discussed. The varying along-gravity effect depends on traction in
the presented model, whereas the cross-gravity additive is taken entirely in
the equations of motion, for any direction and gravity force. We obtain the
conditions for strong convexity and the purely geometric solution to the
problem is given by a new Finsler metric, which belongs to the type of general
$(\alpha, \beta)$-metrics. The proposed model enables us to create a direct
link between the Zermelo navigation problem and the slope-of-a-mountain problem
under the action of a cross gravitational wind. Moreover, the behavior of the
Finslerian indicatrices and time-minimizing trajectories in relation to the
traction coefficient and gravitational wind force are explained and illustrated
by a few examples in dimension two. This also compares the corresponding
solutions on the slippery slopes under various cross- and along-gravity
effects, including the classical Matsumoto's slope-of-a-mountain problem and
Zermelo's navigation.
