Osaka Journal of Mathematics

The duality between singular points and inflection points on wave fronts

Kentaro Saji, Masaaki Umehara, and Kotaro Yamada

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In the previous paper, the authors gave criteria for $A_{k+1}$-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of $2$-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of $M_{-}$. Here $M^{2}$ is a compact orientable 2-manifold, and $M_{-}$ is the open subset of $M^{2}$ where the Hessian of $f$ takes negative values. This is a generalization of Bleecker and Wilson's formula [3] for immersed surfaces in the affine $3$-space.

Article information

Osaka J. Math., Volume 47, Number 2 (2010), 591-607.

First available in Project Euclid: 23 June 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings 53D12: Lagrangian submanifolds; Maslov index
Secondary: 57R35: Differentiable mappings


Saji, Kentaro; Umehara, Masaaki; Yamada, Kotaro. The duality between singular points and inflection points on wave fronts. Osaka J. Math. 47 (2010), no. 2, 591--607.

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