Journal of Singularities
volume 1 (2010), 69-84
Received 6 October 2009. Received in revised form 23 January 2010.
Segre's theorem asserts the following: let a smooth closed simple curve c in S^2 have a non-empty intersection with any closed hemisphere. Then c has at least 4 inflection points. In the paper, we prove two Segre-type theorems. The first one is a version of Segre's theorem for piecewise linear closed curves on S^2. Here we have inflection edges instead of inflection points. Next, we go one dimension higher: we replace S^2 by S^3. Instead of simple curves, we treat immersed saddle surfaces which are homeomorphic to S^2 ("saddle spheres"). We prove that a piecewise linear saddle sphere in S^3 necessarily has inflection or reflex faces. The latter replace inflection points and should be considered as singular phenomena. As an application, we prove that a piecewise linear saddle surface cannot be altered in a neighborhood of its vertex maintaining its saddle property.
Mathematical Subject Classification:
Institute for Informatics and Automation