TL;DR
This paper introduces an affine structure on a cylinder that is almost Zoll, meaning all but one affine geodesic close smoothly, and it is complete and symmetric.
Contribution
It constructs an explicit example of an affine almost Zoll surface on a cylinder with specific geometric properties.
Findings
The affine structure is geodesically complete.
The structure is affine Killing complete.
The structure is affine symmetric.
Abstract
An affine surface is said to be an affine Zoll surface if all affine geodesics close smoothly. It is said to be an affine almost Zoll surface if thru any point, every affine geodesic but one closes smoothly (the exceptional geodesic is said to be alienated as it does not return). We exhibit an affine structure on the cylinder which is almost Zoll. This structure is geodesically complete, affine Killing complete, and affine symmetric.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometric and Algebraic Topology · Geometry and complex manifolds