Non-integrability of cylindric billiards and transitive Lie group actions

A conjecture is formulated and discussed which provides a necessary and suf ficient condition for the ergodicity of cylindric billiards (this conjectur e improves a previous one of the second author). This condition requires th at the action of a Lie-subgroup G of the orthogonal group SO(d) (d being th e dimension of the billiard in question) be transitive on the unit sphere S d(-1). If C-1,..., C-k are the cylindric scatterers of the billiard, then G is generated by the embedded Lie subgroups G(i) of SO(d), where G(i) consi sts of all transformations g is an element of SO(d) of R-d that leave the p oints of the generator subspace of C-i fixed (1 less than or equal to i les s than or equal to k). In this paper we can prove the necessity of our conj ecture and we also formulate some notions related to transitivity. For hard bah systems, we can also show that the transitivity holds in general: for an arbitrary number N greater than or equal to 2 of balls, arbitrary masses m(1),..., m(N) and in arbitrary dimension nu greater than or equal to 2. T his result implies that our conjecture is stronger than the Boltzmann-Sinai ergodic hypothesis for hard ball systems. We also note a somewhat surprisi ng characterization of the positive subspace of the second fundamental form for the evolution of a special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of suffi ciency of an orbit segment.