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

## 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, $\alpha$, on M is contained in a plane, then $\alpha$ 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 $\alpha$(0) = p and $\alpha$'(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 $\alpha$, such that $\alpha$(0) = p and $\alpha$'(0) = v. Since $\alpha$ is a geodesic, by hypothesis it is planar. Then, by theorem, $\alpha$ is also a line of curvature.

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