Intrinsic invariants of cross caps

• Masaru Hasegawa ^[1] ; Atsufumi Honda ^[3] ; Kosuke Naokawa ^[2] ; Masaaki Umehara ^[2] ; Kotaro Yamada ^[2]
  1. [1] Saitama University

     Saitama University

     Minuma-ku, Japón

     
  2. [2] Tokyo Institute of Technology

     Tokyo Institute of Technology

     Japón

     
  3. [3] Miyakonojo National College of Technology

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 20, Nº. 3, 2014, págs. 769-785
• Idioma: inglés
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• Resumen

  • It is classically known that generic smooth maps of RR2RR2 into RR3RR3 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 fstd(u,v)=(u,uv,v2)fstd(u,v)=(u,uv,v2) 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.