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Smooth surfaces with non smooth nullity

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  • Mathematics


My title A FLAT BUT NON-SMOOTHLY RULED SURFACE ANTONIO J. DI SCALA As for everything else, so for a mathematical theory: beauty can be perceived but not explained. A. Cayley a aQuoted in J. R. Newman, The World of Mathematics (New York 1956). Here we discuss an example of flat but not C∞-ruled surface given at page 93 of Differential Geometry Curves-Surfaces-Manifolds by Wolf- gang Ku¨hnel, Second Edition, AMS 2006. A similar, but not explicit example, due to E. Heintze is in 3.9.4 pag.68-39 of Klingenberg’s book “A course in Differential Geometry”, Springer, New York, 1978. The surface is constructed by gluing the red cone C1 with the green cone C2 along a common generatrix (the y-axis). The blue curve c, the generatrix of both cones, is in the xz plane. The vertex of C1 is the point (0, 1, 0) while the vertex of C2 is the point (0,−1, 0). The result- ing surface is C∞ near the origin (0, 0, 0) but is not C∞-ruled around (0, 0, 0). See below for more 3D-plots of the surface. Let c ∈ R3 be a curve in the xz-plane give by c(x) = (x, 0, z(x)) where the function z(x) is C∞, z(0) = 0 and d nz dxn ∣∣ 0 = 0 for n = 1, 2, · · · . We assume also that d 2z(x) dx2 > 0 for x 6= 0. Let C1 be the cone containing c with vertex at the point (0, 1, 0). The tangent plane TOC1 at the origin O is the xy-plane. Since C1 is a C∞ surface near O there exist a C∞-function f1(x, y) such that near O the cone C1 is given by the graph (x, y, f1(x, y)). Date: October 25, 2012. I thank Fabrizio Catanese who asked to me for an example as in Proposition A. I would like to thank Fabio Nicola and Paolo Tilli for discussions about the issue. 1 2 ANTONIO J. DI SCALA A simple calculation shows that near (0, 0) f1(x, y) = z( x 1− y )(1− y) . So for every n,m ∈ N we have (1) ∂n+mf1 ∂xn∂ym ∣∣∣∣ (0,y) = 0 Let now C2 be the cone containing c but with vertex at the point (0,−1, 0). The cone C2 is also given by a graph of a function f2(x, y) near O. By the same argument f2 al

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