  The paper is devoted to the development of the differential geometry of
saccades and saccadic cycles. We recall an interpretation of Donder's and
Listing's law in terms of the Hopf fibration of the $3$-sphere over the
$2$-sphere. In particular, the configuration space of the eye ball (when the
head is fixed) is the 2-dimensional hemisphere $S^+_L$, which is called
Listing's hemisphere. We give three characterizations of saccades: as geodesic
segment $ab$ in the Listing's hemisphere, as the gaze curve and as a piecewise
geodesic curve of the orthogonal group. We study the geometry of saccadic
cycle, which is represented by a geodesic polygon in the Listing hemisphere,
and give necessary and sufficient conditions, when a system of lines through
the center of eye ball is the system of axes of rotation for saccades of the
saccadic cycle, described in terms of world coordinates and retinotopic
coordinates. This gives an approach to the study the visual stability problem.
