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Intrinsic invariants of cross caps |
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| Authors: | Masaru Hasegawa Atsufumi Honda Kosuke Naokawa Masaaki Umehara Kotaro Yamada |
| Affiliation: | 1. Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570, Japan 2. Miyakonojo National College of Technology, 473-1, Yoshiocho, Miyakonojo, Miyazaki, 885-8567, Japan 3. Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo, 152-8551, Japan 4. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1-W8-34, O-okayama, Meguro-ku, Tokyo, 152-8552, Japan |
| Abstract: | It is classically known that generic smooth maps of (varvec{R}^2) into (varvec{R}^3) admit only isolated cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap (f_{mathrm{std }}(u,v)=(u,uv,v^2)) has non-trivial isometric deformations with infinite-dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown. |
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