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November 2020 Isometric deformations of wave fronts at non-degenerate singular points
Atsufumi Honda, Kosuke Naokawa, Masaaki Umehara, Kotaro Yamada
Hiroshima Math. J. 50(3): 269-312 (November 2020). DOI: 10.32917/hmj/1607396490

Abstract

Cuspidal edges and swallowtails are typical non-degenerate singular points on wave fronts in the Euclidean 3-space. Their first fundamental forms belong to a class of positive semi-definite metrics called “Kossowski metrics”. A point where a Kossowski metric is not positive definite is called a singular point or a semi-definite point of the metric. Kossowski proved that real analytic Kossowski metric germs at their non-parabolic singular points (the definition of “non-parabolic singular point” is stated in the introduction here) can be realized as wave front germs (Kossowski’s realization theorem).

On the other hand, in a previous work with K. Saji, the third and the fourth authors introduced the notion of “coherent tangent bundle”. Moreover, the authors, with M. Hasegawa and K. Saji, proved that a Kossowski metric canonically induces an associated coherent tangent bundle.

In this paper, we shall explain Kossowski’s realization theorem from the viewpoint of coherent tangent bundles. Moreover, as refinements of it, we give a criterion that a given Kossowski metric can be realized as the induced metric of a germ of cuspidal edge (resp. swallowtail or cuspidal cross cap). Several applications of these criteria are given. Also, some remaining problems on isometric deformations of singularities of analytic maps are given at the end of this paper.

Funding Statement

The first author is partially supported by the Grant-in-Aid for Young Scientists (B), No. 16K17605. The second author is partially supported by the Grant-in-Aid for Young Scientists (B), No. 17K14197. The third author is partially supported by the Grant-in-Aid for Scientific Research (A) No. 26247005. The fourth author is partially supported by the Grant-in-Aid for Scientific Research (C) No. 26400087.

Citation

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Atsufumi Honda. Kosuke Naokawa. Masaaki Umehara. Kotaro Yamada. "Isometric deformations of wave fronts at non-degenerate singular points." Hiroshima Math. J. 50 (3) 269 - 312, November 2020. https://doi.org/10.32917/hmj/1607396490

Information

Received: 23 May 2019; Revised: 8 April 2020; Published: November 2020
First available in Project Euclid: 8 December 2020

MathSciNet: MR4184262
Digital Object Identifier: 10.32917/hmj/1607396490

Subjects:
Primary: 57R45
Secondary: 53A05

Rights: Copyright © 2020 Hiroshima University, Mathematics Program

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Vol.50 • No. 3 • November 2020
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