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Finsler manifold cubic Weierstrass minimal analog using Descartes Folium

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  • Roger Bagula
    ( Dr. Robert Bryant s work in Finsler space introduced me to this sort of geometry: work in Calabi-Yau surfaces and reading a paper of Dr. Barth s on K3
    Message 1 of 1 , Jun 1, 2008
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      ( Dr. Robert Bryant's work in Finsler space introduced me
      to this sort of geometry: work in Calabi-Yau surfaces
      and reading a paper of Dr. Barth's on K3 surfaces
      inspired this.)
      Here is my idea of a definition of the cubic analog to the Weierstrass
      minimal surface
      in a Finsler cubic "field':
      Let M be a Finsler manifold associated with a cubic
      (M,ds^3) such that they are an orientated Finsler surface.
      Then any point P an element of M is a topological neighborhood
      which can be parametrized in terms of isothermal parameters.
      Call N: M->S^2 it's Descartes map.
      Denote H: M->Real as the mean curvature associated with map N
      such that the analog:
      deltaX=3*H*N
      Then the (M,ds^3) produces Harmonic 1-forms on M of
      Sum[dX(i)^3,{i,1,3}]=0
      In terms like those traditional to Weierstrass minimal surfaces
      in this cubic analog:
      phi1(f,g)=3*f*g/(1+g^3)
      phi2(f,g)=3*f*g^/(1+g^3)
      phi3(f,g)=-3*f*g/(1+g^3)^(2/3)
      Mathematica:
      phi1[f_, g_] = 3*f*g/(1 + g^3)
      phi2[f_, g_] = 3*f*g^2/(1 + g^3)
      phi3[f_, g_] = -3*f*g/(1 + g^3)^(2/3)
      FullSimplify[phi1[f, g]^3 + phi2[f, g]^3 + phi3[f, g]^3]

      In the simple f=1; g=t folium case
      I have integrated a path on the Manifold M in Mathematica.
      In three parts the note book is:
      Part one folium 3d projected on triazial "tube" cyclinder:
      Clear[x, y, x, t, f, g, h, s, g0]
      x = 3*t/(1 + t^3)
      y = 3*t^2/(1 + t^3)
      z = -3*t/(1 + t^3)^(2/3)
      FullSimplify[x^3 + y^3 + z^3]
      ParametricPlot3D[{x, y,
      Re[z]}, {t, -3, 3}, Axes -> False, Boxed -> False, PlotPoints -> 1000]
      g0 = ParametricPlot3D[{x*Cos[p], y*Cos[p + 2*Pi/3], Re[z]*Cos[p +
      4*Pi/3]}, {p, 0, Pi}, {t, 0.01, 4}, Axes -> False, Boxed -> False,
      PlotPoints -> {40, 40}]
      ga = ParametricPlot3D[{x*
      Cos[p], y*Cos[p + 2*Pi/3],
      Re[z]*Cos[p + 4*Pi/3]}, {p, 0, Pi}, {t, -0.99, -0.01}, Axes ->
      False, Boxed -> False, PlotPoints -> {40, 40}]
      Show[g0, ViewPoint -> {-0.061, 3.381, - 0.127}]
      Show[{g0, ga}, ViewPoint -> {-0.061, 3.381, - 0.127}]

      Part two (with Mathematica specific characters)
      the intgrations:
      \!\(f[s_] = Integrate[x, {t, \(-3\), s}]\n
      f1[s_] = FullSimplify[3\ \((3 + s)\)*\((\(2\ \[ImaginaryI]\
      ? + 2\ \@3\ ArcTan[7\/\@3] -
      2\ \@3\ ArcTan[\(1 - 2\ s\)\/\@3] - Log[13\/4] - 2\ Log[1 + s] + Log[1 \
      - s + s\^2]\)\/\(6\ \((3 + s)\)\))\)]\n
      g[s_] = Integrate[y, {t, \(-3\), s}]\n
      g1[s_] = FullSimplify[3\ \((3 + s)\)*\((\(\(-\[ImaginaryI]\)\ ? - Log[26] + \
      Log[1 + s\^3]\)\/\(3\ \((3 + s)\)\))\)]\n
      h[s_] = Integrate[z, {t, \(-3\), s}]\n
      h1[s_] = FullSimplify[3\ \((3 +
      s)\)*\((\(9\ \((\(-1\))\)\^\(2/3\)\ Hypergeometric2F1[
      2\/3, 2\/3, 5\/3, 27] + s\ \((\(-\(s\^3\/\((1 +
      s\^3)\)\^2\)\))\)\^\(1/3\)\ \((1 + s\^3)\)\^\(2/3\)\ Hypergeometric2F1[
      2\/3, 2\/3, 5\/3, \(-s\^3\)]\)\/\(2\ \((3 + s)\)\))\)]\)

      Part three the poltting of the new surface:
      ParametricPlot3D[{Re[f1[s]], Re[g1[s]], Re[h1[s]]}, {s, -10, 10}, Axes -> \
      False, Boxed -> False, PlotPoints -> 1000]
      gb = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*
      Pi/3], Re[h1[s]]*Cos[p + 4*Pi/3]}, {p, 0,
      Pi}, {s, 0.01, 4}, Axes -> False, Boxed -> False, PlotPoints -> {40, 40}]
      gc = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/3], Re[h1[
      s]]*Cos[p + 4*Pi/3]}, {p, 0,
      Pi}, {s, -0.99, -0.01}, Axes -> False, Boxed -> False, PlotPoints \
      -> {40, 40}]
      gd = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/
      3], Re[h1[s]]*Cos[p + 4*Pi/3]}, {p, -Pi, 0}, {s, 0.01, 4},
      Axes -> False, Boxed -> False, PlotPoints -> {40, 40}]
      ge = ParametricPlot3D[{Re[f1[s]]*Cos[p], Re[g1[s]]*Cos[p + 2*Pi/3],
      Re[h1[s]]*Cos[p + 4*Pi/3]}, {
      p, -Pi, 0}, {s, -0.99, -0.01}, Axes -> False, Boxed -> False,
      PlotPoints -> {40, 40}]
      Show[{gb, gc, gd, ge}, ViewPoint -> {-0.061, 3.381, - 0.127}]
      Show[{gb, gc, gd, ge}]

      Picture of the surface is attached.

      This plain simple surface is just an indication of
      other possible surfaces by conformal mapping functions of (f,g)
      onto the Descartes map and integrating.

      That is a Finsler type manifold with a 1/r^3 force.

      It gives a strange twisted torus of a negative curvature like

      a pseudosphere with a hole in it that has a perpendicular

      pseudosphere to it.

      This is the K3 like case of a Finsler manifold with the

      folium loop in it.

      It is probably closer to what a real Calabi-Yau / Null-Ricci

      for these non-Riemannian geometries would look like.

      Respectfully, Roger L. Bagula
      11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :http://www.geocities.com/rlbagulatftn/Index.html
      alternative email: rlbagula@...
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