2015/09/16 by Solomon, Bruce
#53A05 #53C42 #53C45 #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1509.04967
We prove: If a complete connected smooth surface M in euclidean 3-space has general position, intersects some plane along a clean figure-8 (a loop with total curvature zero) and all compact intersections with planes have central symmetry, then M is a (geometric) cylinder over some central figure-8. On the way, we establish interesting facts about centrally symmetric loops in the plane; for instance, a clean loop with even rotation number 2k can never be central unless it passes through its center exactly twice and k=0.