Singularities of piecewise linear saddle spheres on S^3
Journal of Singularities, Tome 1 (2010), pp. 69-84
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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.
@article{10_5427_jsing_2010_1e,
author = {Gaiane Panina},
title = {Singularities of piecewise linear saddle spheres on {S^3}},
journal = {Journal of Singularities},
pages = {69--84},
publisher = {mathdoc},
volume = {1},
year = {2010},
doi = {10.5427/jsing.2010.1e},
url = {http://geodesic.mathdoc.fr/articles/10.5427/jsing.2010.1e/}
}
Gaiane Panina. Singularities of piecewise linear saddle spheres on S^3. Journal of Singularities, Tome 1 (2010), pp. 69-84. doi: 10.5427/jsing.2010.1e
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