Abstract
We generalize the definition of a flecnode on a surface in ℝ3 to a definition for a general immersed manifold in Euclidean space. Instead of considering a flecnode as a point on the manifold, we consider it as a pair of a normal vector and a tangent vector, called the flecnodal pair. The structure of this set is considered, as well as its connection to binormals and A 3 singularities in the set of height functions. The specific case of a surface immersed in ℝ4 is studied in more detail, with the generic singularities of the flecnodal normals and the flecnodal tangents classified. Finally, the connection between the flecnodals and bitangencies are studied, especially in the case where the dimension of the manifold equals the codimension.
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References
T. Banchoff, T. Gaffney, and C. McCrory. Cusps of Gauss Mappings, volume 55 of Research Notes in Mathematics. Pittman, 1982.
J.W. Bruce and A.C. Nogueira. Surfaces in ℝ4 and duality. Quart. J. Math., 2(49):433–443, 1998.
J.W. Bruce and F. Tari. Families of surfaces in ℝ4. Proc. Edinburgh Math. Soc., 45:181–203, 2002.
D. Dreibelbis. A bitangency theorem for surfaces in four-dimensional Euclidean space. Q. J. Math., 52:137–160, 2001.
D. Dreibelbis. Singularities of the guass map and the binormal surface. Adv. Geom., 3:453–468, 2003.
D. Dreibelbis. Birth of bitangencies in a family of surfaces in ℝ4. Submitted, Differential Geometry and its Applications, 2004.
D. Dreibelbis. Invariance of the diagonal contribution in a bitangency formula. In Real and Complex Singularities, volume 354 of Contemporary Mathematics, pages 45–55. American Mathematical Society, 2004.
R.A. Garcia, D.K.H. Mochida, M.C. Romero Fuster, and M.A.S. Ruas. Inflection points and topology of surfaces in 4-space. Trans. Amer. Math. Soc., 352(7):3029–3043, 2000.
D. Mochida, M.C. Romero Fuster, and M. Ruas. The geometry of surfaces in 4-space from a contact viewpoint. Geom. Dedicata, 54:323–332, 1995.
I.R. Porteous. Geometric Differentiation. Cambridge University Press, 1994.
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Dreibelbis, D. (2006). The Geometry of Flecnodal Pairs. In: Brasselet, JP., Ruas, M.A.S. (eds) Real and Complex Singularities. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7776-2_9
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DOI: https://doi.org/10.1007/978-3-7643-7776-2_9
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7775-5
Online ISBN: 978-3-7643-7776-2
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