  It is shown that the classical motion of massive particles in hyperbolic
spaces $H^D$ has a bounded character in $D-1$ coordinates. Studying the Dirac
equation, it is found that a bounded character of the classical motion
corresponds to the effective dimensional reduction $D+1 \to 1+1$ for fermions
in the infrared region in the quantum problem. This effective dimensional
reduction leads to the zero critical value of coupling constant for dynamical
symmetry breaking in hyperbolic spaces.
