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
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
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