  A system of $N$ rotators is investigated with a constraint given by a
condition of vanishing sum of the cosines of the rotation angles. Equations of
the dynamics are formulated and results of numerical simulation for the cases
$N$=3, 4, and 5 are presented relating to the geodesic flows on a
two-dimensional, three-dimensional, and four-dimensional manifold,
respectively, in a compact region (due to the periodicity of the configuration
space in angular variables). It is shown that a system of three rotators
demonstrates chaos, characterized by one positive Lyapunov exponent, and for
systems of four and five elements there are, respectively, two and three
positive exponents (`hyperchaos'). An algorithm has been implemented that
allows calculating the sectional curvature of a manifold in the course of
numerical simulation of the dynamics at points of a trajectory. In the case of
$N$=3, curvature of the two-dimensional manifold is negative (except for a
finite number of points where it is zero), and Anosov's geodesic flow is
realized. For $N$=4 and 5, the computations show that the condition of negative
sectional curvature is not fulfilled. Also the methodology is explained and
applied for testing hyperbolicity based on numerical analysis of the angles
between the subspaces of small perturbation vectors; in the case of $N$=3, the
hyperbolicity is confirmed, and for $N$=4 and 5 the hyperbolicity does not take
place.
