1996/04/01 by W. L. Marar, Ton Marar, F. Tari +1 · 1 citation
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.1017/s030500410007434x
openalex publication_date 1996/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26
In this paper we investigate the geometry of simple germs of co-rank 1 maps from ℝ 3 to ℝ 3 . 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 ℝ 3 with an initial velocity distribution v = v ( x ) (and a positive density distribution), the initial motion of particles defines a time-dependent map g t : ℝ 3 → ℝ 3 given by g t ( x ) = x + tv ( x ). At some time t singularities occur and the critical values of g i 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 : ℝ 3 , 0 → ℝ 3 , 0.