4 Answer summary and new question.

The geodesics on a cylinder (a cylinder infinite in both directions) are either (1) simple (non-self-intersecting) closed geodesics, or (2) simple infinitely long geodesics (infinite in both directions).

(Image from John Oprea.)

Q

Q1. Are there any other surfaces in $\mathbb{R}^3$ all of whose geodesics are either class (1) or class (2) above (and there are geodesics in both classes)?

In other words, does this classification of geodesics characterize the cylinder? Thanks for your thoughts!

Answered. Anton Petrunin's example shows that the answer to my question above is No. However, Paul Reynolds raised a possibly more interesting variant of my question:

Q2. Are there any other surfaces in $\mathbb{R}^3$, topologically distinct from a cylinder, all of whose geodesics are either class (1) or class (2) above (and there are geodesics in both classes)?

In other words, does this classification of geodesics determine that such a surface must have the topology of a cylinder?

3 edited body

The geodesics on a cylinder (a cylinder infinite in both directions) are either (1) simple (non-self-intersecting) closed geodesics, or (2) simple infinitely long geodesics (infinite in both directions).

(Image from John OperaOprea.)

Q. Are there any other surfaces in $\mathbb{R}^3$ all of whose geodesics are either class (1) or class (2) above (and there are geodesics in both classes)?

In other words, does this classification of geodesics characterize the cylinder? Thanks for your thoughts!

Q. Are there any other surfaces in $\mathbb{R}^3$ all of whose geodesics are either class (1) or class (2) above (and there are geodesics in both classes)?