At the risk of asking an uninformed question...

Imagine an ant on a two-dimensional surface embedded in 3-space. The ant is placed at an arbitrary point on the surface with random orientation. Once placed and oriented, the ant will walk along the curvature of the surface.

Are there examples of such 2D surfaces where we are guaranteed the ant will never return to its (arbitrary) starting position?

At the risk of asking an uninformed question...

Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once placed and oriented, the ant will walk along a local geodesic path.

Are there examples of such 2D surfaces where we are guaranteed the ant will never return to some starting position?

(Thanks to Henry Wilton for requesting clarification as per his comment below.)