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Frame and direction mappings for surfaces in ℝ3

Part of: Classical differential geometry Theory of singularities and catastrophe theory General theory in ordinary differential equations

Published online by Cambridge University Press:  26 January 2019

J. W. Bruce  and
F. Tari
J. W. Bruce
Affiliation:
Department of Mathematical Sciences, University of Liverpool, Mathematical Sciences Building, Liverpool L69 7ZL, UK (billbrucesingular@gmail.com)
F. Tari
Affiliation:
Instituto de Ciências Matemáticas e de Computacão - USP, Avenida Trabalhador são-carlense, 400 - Centro, CEP: 13566-590, São Carlos, SP, Brazil (faridtari@icmc.usp.br)
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Abstract

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations.

Keywords

Framesprincipal directions and curvesasymptotic directions and curvessingularitiesfirst order differential equationsdivergent dagrams

MSC classification

Primary: 53A05: Surfaces in Euclidean space
Secondary: 58K05: Critical points of functions and mappings 34A09: Implicit equations, differential-algebraic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 3 , June 2019 , pp. 795 - 830
Copyright
Copyright © Royal Society of Edinburgh 2019 

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References

1Arnol'd, V. I., Guseĭn-Zade, S. M. and Varchenko, A. N.. Singularities of differentiable maps, vol. I, The classification of critical points, caustics and wave fronts. Monographs in Mathematics, 82 (Boston, MA: Birkhäuser Boston, Inc., 1985).Google Scholar
2Banchoff, T. F., Gaffney, T. J. and McCrory, C.. Cusps of Gauss Mappings. Research Notes in Mathematics,vol. 55 (London: Pitman, 1982).Google Scholar
3Bruce, J. W.. A note on first order differential equations of degree greater than one and wavefront evolution. Bull. London Math. Soc. 16 (1984a), 139–144.Google Scholar
4Bruce, J. W. and Tari, F.. Extrema of principal curvature and symmetry. Proc. Edinburgh Math. Soc. 39 (1996), 397–402.Google Scholar
5Bruce, J. W. and Tari, F.. Duality and implicit differential equations. Nonlinearity 13 (2000), 791–812.Google Scholar
6Bruce, J. W. and Wilkinson, T. C.. Folding maps and focal sets. Lecture Notes in Math. 1462,pp. 63–72 (Berlin: Springer, 1991).Google Scholar
7Davydov, A. A., Ishikawa, G., Izumiya, S. and Sun, W.-Z.. Generic singularities of implicit systems of first order differential equations on the plane. Jpn. J. Math. 3 (2008), 93–119.Google Scholar
8Dufour, J.-P.. Familles de courbes planes différentiables. Topology 22 (1983), 449–474.Google Scholar
9Dufour, J.-P. and Kurokawa, Y.. Topological rigidity for certain families of differentiable plane curves. J. London Math. Soc. 65 (2002), 223–242.Google Scholar
10Hayakawa, A., Ishikawa, G., Izumiya, S. and Yamaguchi, K.. Classification of generic integral diagrams and first order ordinary differential equations. Int. J. Math. 5 (1994), 447–489.Google Scholar
11Izumiya, S., Romero–Fuster, M. C., Ruas, M. A. S. and Tari, F.. Differential geometry from a singularity theory viewpoint (Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2016).Google Scholar
12Mond, D. M. Q.. On the classification of germs of maps from ℝ2 to ℝ3. Proc. London Math. Soc. 50 (1985), 333–369.Google Scholar
13Morris, R. J.. The sub-parabolic lines of a surface. The mathematics of surfaces, VI, Inst. Math. Appl. Conf. Ser. New Ser.,vol. 58, pp. 79–102 (Uxbridge: Oxford Univ. Press, 1994).Google Scholar
14O'Neill, B.. Elementary differential geometry (New York: Academic Press, 1966).Google Scholar
15Rieger, J. H.. Families of maps from the plane to the plane. J. London Math. Soc. 36 (1987), 351–369.Google Scholar
16Saji, K.. Criteria for singularities of smooth maps from the plane into the plane and their applications. Hiroshima Math. J. 40 (2010), 229–239.Google Scholar
17Wall, C. T. C.. Finite determinacy of smooth map-germs. Bull. London Math. Soc. 13 (1981), 481–539.Google Scholar
18Wilkinson, T. C.. The geometry of folding maps. PhD Thesis, University of Newcastle-upon-Tyne (1991).Google Scholar