Symmetries of cross caps
Atsufumi Honda, Kosuke Naokawa, Kentaro Saji, Masaaki Umehara and, Kotaro Yamada
TL;DR
This paper investigates the uniqueness of Bruce-West's normal form for cross caps in Euclidean 3-space, introduces new invariants from associated functions, and classifies symmetries of cross cap singular points.
Contribution
It establishes the uniqueness of the normal form for cross caps and introduces new invariants that enable classification of their symmetries.
Findings
Normal form for cross caps is unique up to certain transformations.
New invariants are derived from functions associated with the normal form.
Symmetries of cross caps are classified based on these invariants.
Abstract
It is well-known that cross caps on surfaces in the Euclidean 3-space can be expressed in Bruce-West's normal form, which is a special local coordinate system centered at the singular point. In this paper, we show a certain kind of uniqueness of such a coordinate system. In particular, the functions associated with this coordinate system produce new invariants on cross cap singular points. Using them, we classify the possible symmetries on cross caps.
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