BEHAVIOR OF CORANK-ONE SINGULAR POINTS ON WAVE FRONTS

概要

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Let $ M^2 $ be an oriented 2-manifold and $ f \colon M^2 \rightarrow R^3 $ a $ C^\infty $-map. A point $ p \in M^2 $ is called a $ \mathnormal{singular point} $ if $ f $ is not an immersion at $ p $. The map $ f $ is called a front (or $ \mathnormal{wave front} $), if there exists a unit $ C^\infty $-vector field $ \nu $ such that the image of each tangent vector $ df(X) (X \in TM^2) $ is perpendicular to $ \nu $, and the pair $ (f, ν) $ gives an immersion into $ R^3 \times S^2 $. In a previous paper, we gave an intrinsic formulation of wave fronts in $ R^3 $. In this paper, we investigate the behavior of cuspidal edges near corank-one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation.
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