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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

REFERENCES

[1]Arnold, V. I.. Normal forms for functions near degenerate critical points, the Weyl groups of An, Dn, En and Legendrean singularities. Funt. Anal, and its Appl. 6 (1972), 254–272.CrossRefGoogle Scholar
[2]Arnold, V. I.. Wavefront evolution and equivariant Morse lemma. Comm. Pure and App. Math. 29 (1976), 557–582.CrossRefGoogle Scholar
[3]Arnold, V. I.. Catastrophe theory, 3rd edition (Springer-Verlag, 1992).CrossRefGoogle Scholar
[4]Bruce, J. W.. A classification of 1-parameter families of map-germs ℝ3, 0 → ℝ3, 0 with application to condensation problems. J. London Math. Soc. (2) 33 (1986), 375–384.CrossRefGoogle Scholar
[5]Bruce, J. W. and Duplessis, A. A.. Complete transversals (in preparation).Google Scholar
[6]Bruce, J. W. and Marar, W. L.. Images and varieties (preprint) (Liverpool University 1993).Google Scholar
[7]Damon, J. and Mond, D., A-codimension and vanishing topology of discriminants. Invent. Math. 106 (1991), 217–242.CrossRefGoogle Scholar
[8]Dufour, J. P. and Jean, P.. Familles de surfaces differentiables. J. London Math. Soc. (2) 42 (1990), 175–192.CrossRefGoogle Scholar
[9]Duplessis, A. A.. On the determinacy of smooth map-germs. Invenliones Math. 58 (1980), 107–160.CrossRefGoogle Scholar
[10]Duplessis, A. A.. Unfoldings and A-equivalence (in preparation).Google Scholar
[11]Duplessis, A. A., Gaffney, T. and Wilson, L.. Map-germs determined by their discriminants (preprint) (Aarhus University, 1993).Google Scholar
[12]Duplessis, A. A. and Tari, F., A-classification of map-germs (in preparation).Google Scholar
[13]Gorynunov, V. V.. Semi-simplicial resolutions and homology of images and discriminants of mappings, to appear in Proc. London Math. Soc.Google Scholar
[14]Kirk, N. P.. Computational aspects of singularity theory, Ph.D. Thesis (Liverpool, 1993).Google Scholar
[15]Trano, Lé Dung. Calculation of the Milnor number of isolated singularity of complete intersection. Funk. Anal, i Ego Pril. 8 (1974), 45–49.Google Scholar
[16]Marar, W. L. and Mond, D.. Multiple point schemes for corank 1 maps. J. London Math. Soc. (2) 39 (1989), 553–567.CrossRefGoogle Scholar
[17]Marar, W. L., Montaldi, J. and Ruas, M. A. S.. Multiplicities of zero-schemes in quasi-homogeneous singularities (in preparation).Google Scholar
[18]Mond, D.. On the classification of maps from ℝ2, 0→ ℝ3, 0. Proc. London Math. Soc. 50 (1985), 333–369.CrossRefGoogle Scholar
[19]Mond, D.. Some remarks on the geometry and classification of germs of maps from surfaces to 3-space. Topology 26 (1987), 361–383.CrossRefGoogle Scholar
[20]Mond, D.. Singularities of the tangent developable surface of a space curve. Quart. J. Math. Oxford 40 (1989), 79–81.CrossRefGoogle Scholar
[21]Mond, D. and Pellikaan, R.. Fitting ideals for analytic maps. Lecture Notes in Math. 1414 (Springer-Verlag, 1987), pp. 107–161.Google Scholar