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February 2019 Isometric realization of cross caps as formal power series and its applications
Atsufumi HONDA, Kosuke NAOKAWA, Masaaki UMEHARA, Kotaro YAMADA
Hokkaido Math. J. 48(1): 1-44 (February 2019). DOI: 10.14492/hokmj/1550480642

Abstract

Two cross caps in Euclidean 3-space are said to be infinitesimally isometric if their Taylor expansions of the first fundamental forms coincide by taking a local coordinate system. For a given $C^\infty$ cross cap $f$, we give a method to find all cross caps which are infinitesimally isomeric to $f$. More generally, we show that for a given $C^{\infty}$ metric with singularity having certain properties like as induced metrics of cross caps (called a Whitney metric), there exists locally a $C^\infty$ cross cap infinitesimally isometric to the given one. Moreover, the Taylor expansion of such a realization is uniquely determined by a given $C^{\infty}$ function with a certain property (called characteristic function). As an application, we give a countable family of intrinsic invariants of cross caps which recognizes infinitesimal isometry classes completely.

Citation

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Atsufumi HONDA. Kosuke NAOKAWA. Masaaki UMEHARA. Kotaro YAMADA. "Isometric realization of cross caps as formal power series and its applications." Hokkaido Math. J. 48 (1) 1 - 44, February 2019. https://doi.org/10.14492/hokmj/1550480642

Information

Published: February 2019
First available in Project Euclid: 18 February 2019

zbMATH: 07055593
MathSciNet: MR3914167
Digital Object Identifier: 10.14492/hokmj/1550480642

Subjects:
Primary: 57R45
Secondary: 53A05

Rights: Copyright © 2019 Hokkaido University, Department of Mathematics

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Vol.48 • No. 1 • February 2019
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