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].
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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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