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

Published online by Cambridge University Press:  24 October 2008

W. L. Marar  and
F. Tari
W. L. Marar
Affiliation:
Instituto de Ciências, Matemáticas de São Carlos, Universidade de São Paulo, Caixa Postal 668, 13560 São Carlos (SP), Brazil
F. Tari
Affiliation:
Silsoe Research Institute, Mathematics and Decision Systems Group, Wrest Park, Silsoe, Bedford MK45 4HS
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Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 119 , Issue 3 , April 1996 , pp. 469 - 481
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

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