Hokkaido Mathematical Journal

Isometric realization of cross caps as formal power series and its applications

Atsufumi HONDA, Kosuke NAOKAWA, Masaaki UMEHARA, and Kotaro YAMADA

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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.

Article information

Hokkaido Math. J., Volume 48, Number 1 (2019), 1-44.

First available in Project Euclid: 18 February 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R45: Singularities of differentiable mappings
Secondary: 53A05: Surfaces in Euclidean space

cross cap Whitney umbrella positive semi-definite metric isometric deformation intrinsic invariant


HONDA, Atsufumi; NAOKAWA, Kosuke; UMEHARA, Masaaki; YAMADA, Kotaro. Isometric realization of cross caps as formal power series and its applications. Hokkaido Math. J. 48 (2019), no. 1, 1--44. doi:10.14492/hokmj/1550480642. https://projecteuclid.org/euclid.hokmj/1550480642

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