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

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Homework Statement



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.

Homework 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.

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?
 

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