On the geometry of simple germs of co-rank 1 maps from ℝ3 to ℝ3

In this paper we investigate the geometry of simple germs of co-rank 1 maps from ℝ3 to ℝ3. Those of co-dimension 1 have already been dealt with by several authors. In [2], V. I. Arnold considered the problem of evolution of galaxies. For a medium of non-interacting particles in ℝ3 with an initial velocity distribution v = v(x) (and a positive density distribution), the initial motion of particles defines a time-dependent map gt: ℝ3 → ℝ3 given by gt(x) = x + tv(x). At some time t singularities occur and the critical values of gi correspond to points of condensation of particles. Arnold assumed the vector field v is a gradient, that is v = ∇S, for some potential S. J. W. Bruce generalized these results in [4] by dropping the assumption on the velocity distribution and studied generic 1-parameter families of map germs F: ℝ3, 0 → ℝ3, 0.