Abstract
This is a survey article on isometric deformations of surfaces with singularities. At the end of this paper, the author introduces a new problem on isometric deformations of cross cap singularities.
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Notes
- 1.
The precise proof of this theorem is written in [11].
- 2.
The standard cuspidal edge as in Fig. 1, left, has an identically vanishing limiting normal curvature. In fact, it is a developable surface.
- 3.
One cannot replace the condition \(\vert \kappa _{s}(t)\vert <\tilde{\kappa } (t)\) by \(\vert \kappa _{s}(t)\vert \leq \tilde{\kappa } (t)\) , see [8].
- 4.
Since the product \(\vert \kappa _{\nu }\kappa _{c}\vert \) is intrinsic, \(\kappa _{c}\)is an extrinsic invariant.
- 5.
\(df_{p}(T_{p}U^{2})\) i a 1-dimensional vector space, which is called the tangential direction of f.
- 6.
Such kinds of cross caps are called normal cross caps. A geometric meaning of normal cross caps are given in [3].
References
Fukui, T., Hasegawa, M.: Fronts of Whitney umbrella—a differential geometric approach via blowing up. J. Singul. 4, 35–67 (2012)
Hasegawa, M., Honda, A., Naokawa, K., Saji, K., Umehara, M., Yamada, K.: Intrinsic properties of singularities of surfaces, preprint. arXiv:1409.0281
Hasegawa, M., Honda, A., Naokawa, K., Umehara, M., Yamada, K.: Intrinsic invariants of cross caps. Selecta Mathematica 20, 769–785 (2014)
Honda, A., Naokawa, K., Umehara, M., Yamada, K.: In: Direct correspondence at a meeting at March (2013)
Kossowski, M.: Realizing a singular first fundamental form as a nonimmersed surface in Euclidean 3-space. J. Geom. 81, 101–113 (2004)
Martins, L.F., Saji, K.: Geometric invariants of cuspidal edges, preprint (2013) Available from www.ibilce.unesp.br/Home/Departamentos/%Matematica/Singularidades/martins-saji-geometric.pdf
Martins, L.F., Saji, K., Umehara, M., Yamada, K.: Behavior of Gaussian curvature around non-degenerate singular points on wave fronts, preprint, arXiv:1308.2136 (2013)
Naokawa, K., Umehara, M., Yamada, K.: Isometric deformations of cuspidal edges, preprint. arXiv:1408.4243
Saji, K., Umehara, M., Yamada,K.: The geometry of fronts. Ann. Math. 169, 491–529 (2009)
Shiba, S., Umehara, M.: The behavior of curvature functions at cusps and inflection points. Differ. Geom. Appl. 30, 285–299 (2012)
Spivak, M.: A comprehensive Introduction to Differential Geometry V. Pelish Inc., Houston (1999)
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Umehara, M. (2014). Isometric Deformations of Surfaces with Singularities. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds) Real and Complex Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 106. Springer, Tokyo. https://doi.org/10.1007/978-4-431-55215-4_12
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