Intrinsic invariants of cross caps


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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The authors thank Shyuichi Izumiya, Toshizumi Fukui and Wayne Rossman for valuable comments. The fourth author thanks Huili Liu for fruitful discussions on this subject at 8th Geometry Conference for Friendship of China and Japan at Chengdu.

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Correspondence to Masaaki Umehara.

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The second and third authors were partly supported by the Grant-in-Aid for JSPS Fellows. The fourth and fifth authors were partially supported by Grant-in-Aid for Scientific Research (A) No. 22244006, and Scientific Research (B) No. 21340016, respectively, from the Japan Society for the Promotion of Science.

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Hasegawa, M., Honda, A., Naokawa, K. et al. Intrinsic invariants of cross caps. Sel. Math. New Ser. 20, 769–785 (2014).

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  • Cross cap
  • Curvature
  • Isometric deformation

Mathematics Subject Classification (2010)

  • Primary 57R45
  • Secondary 53A05