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

[M512]

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 T_pM. 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 T_pM, 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?

 

[mighty2000]

To show that p is umbilic, we can use the first and second fundamental forms of the surface M. Since \alpha is a line of curvature, its tangent vector \alpha' is perpendicular to the normal vector N at any point. This means that the normal curvature at p is equal to the principal curvature in the direction of \alpha'.

Since every geodesic on M is a plane curve, the principal curvature in the direction of \alpha' is constant for all points on M. This means that the normal curvature at p is also constant. Therefore, p is an umbilic point.

Since this is true for any point p on M, we can conclude that M consists entirely of umbilic points. By the theorem, this means that M is contained in either a plane or a sphere.