Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?