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Differential Geometry! Surface with planar geodesics is always a sphere or plane!

  1. Nov 16, 2011 #1
    1. The problem statement, all variables and given/known data

    Show that if M is a surface such that every geodesic is a plane curve, then M is a part of a plane or a sphere.

    2. Relevant equations

    - If a geodesic, [itex]\alpha[/itex], on M is contained in a plane, then [itex]\alpha[/itex] is also a line of curvature.
    - Let p be any point on a surface M and let a vector v be an element of TpM. Then there is a unique geodesic alpha such that [itex]\alpha[/itex](0) = p and [itex]\alpha[/itex]'(0) = v.
    - A surface M consisting entirely of umbilic points is contained in either a plane or a sphere.

    3. The attempt at a solution

    Let p be any point in M. Then for any vector v of TpM, there is a geodesic [itex]\alpha[/itex], such that [itex]\alpha[/itex](0) = p and [itex]\alpha[/itex]'(0) = v. Since [itex]\alpha[/itex] is a geodesic, by hypothesis it is planar. Then, by theorem, [itex]\alpha[/itex] is also a line of curvature.

    Now I'm having trouble showing that p is umbilic. Help?
     
  2. jcsd
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