Commentarii Mathematici Helvetici


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Volume 87, Issue 2, 2012, pp. 243–270
DOI: 10.4171/CMH/253

Published online: 2012-04-17

Surfaces with central convex cross-sections

Bruce Solomon[1]

(1) Indiana University, Bloomington, USA

Say that a surface in $S\subset\mathbb{R}^{3}$ has the central plane oval property, or cpo, if

  • $S$ meets some affine plane transversally along an oval, and
  • Every such transverse plane oval on $S$ has central symmetry.
We show that a complete, connected $C^{2}$ surface with cpo must either be a cylinder over a central oval, or else quadric. We apply this to deduce that a complete $C^{2}$ surface containing a transverse plane oval but no skewloop must be cylindrical or quadric.

Keywords: Quadric surface, oval, central symmetry, skewloop

Solomon Bruce: Surfaces with central convex cross-sections. Comment. Math. Helv. 87 (2012), 243-270. doi: 10.4171/CMH/253