Framed Surfaces in the Euclidean Space

Arnol’d, V.I.: Singularities of Caustics and Wave Fronts. Mathematics and Its Applications, vol. 62. Kluwer, Dordrecht (1990) Arnol’d, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Basel (1986) Bishop, R.L.: There is more than one way to frame a curve. Am. Math. Mon. 82, 246–251 (1975) Bruce, J.W., Giblin, P.J.: Curves and Singularities. A Geometrical Introduction to Singularity Theory., 2nd edn. Cambridge University Press, Cambridge (1992) Fujimori, S., Saji, K., Umehara, M., Yamada, K.: Singularities of maximal surfaces. Math. Z. 259, 827–848 (2008) Fukui, T.: Local differential geometry of cuspidal edge and swallowtail (preprint) (2017) Fukui, T., Hasegawa, M.: Fronts of Whitney umbrella—a differential geometric approach via blowing up. J. Singul. 4, 35–67 (2012) Fukunaga, T., Takahashi, M.: Existence and uniqueness for Legendre curves. J. Geom. 104, 297–307 (2013) Fukunaga, T., Takahashi, M.: Evolutes of fronts in the Euclidean plane. J. Singul. 10, 92–107 (2014) Fukunaga, T., Takahashi, M.: Framed surfaces and one-parameter families of framed curves (preprint) (2018) Gray, A., Abbena, E., Salamon, S.: Modern Differential Geometry of Curves and Surfaces with Mathematica. Studies in Advanced Mathematics., 3rd edn. Chapman and Hall/CRC, Boca Raton (2006) Honda, S., Takahashi, M.: Framed curves in the Euclidean space. Adv. Geom. 16, 265–276 (2016) Ishikawa, G.: Zariski’s Moduli Problem for Plane Branches and the Classification of Legendre Curve Singularities. Real and Complex Singularities, pp. 56–84. World Sci. Publ, Hackensack (2007) Ishikawa, G.: Singularities of Curves and Surfaces in Various Geometric Problems. CAS Lecture Notes, vol. 10, Exact Sciences (2015) Izumiya, S., Saji, K.: The mandala of Legendrian dualities for pseudo-spheres in Lorentz–Minkowski space and “flat” spacelike surfaces. J. Singul. 2, 92–127 (2010) Izumiya, S., Romero-Fuster, M.C., Ruas, M.A.S., Tari, F.: Differential Geometry from a Singularity Theory Viewpoint. World Scientific, New Jersey (2015) Ivey, T.A., Landsberg, J.M.: Cartan for Beginners. Differential Geometry via Moving Frames and Exterior Differential Systems, 2nd edn. Graduate Studies in Mathematics, vol. 175. American Mathematical Society, Providence (2016) Kokubu, M., Rossman, W., Saji, K., Umehara, M., Yamada, K.: Singularities of flat fronts in hyperbolic space. Pac. J. Math. 221, 303–351 (2005) Martins, L., Nuño-Ballesteros, J.J.: Contact properties of surfaces in \({\mathbb{R}}^3\) with corank \(1\) singularities. Tohoku Math. J. (2) 67, 105–124 (2015) Martins, L., Saji, K.: Geometric invariants of cuspidal edges. Can. J. Math. 68, 445–462 (2016) Martins, L., Saji, K., Umehara, M., Yamada, K.: Behavior of Gaussian Curvature and Mean Curvature Near Non-degenerate Singular Points on Wave Fronts. Geometry and Topology of Manifolds, Springer Proc. Math. Stat, vol. 154, pp. 247–281 (2016) Oset Sinha, R., Saji, K.: On the geometry of the cuspidal cross-cap preprint) (2017). arXiv:1706.02074v1 Oset Sinha, R., Tari, F.: Projections of surfaces in \({\mathbb{R}}^4\) to \({\mathbb{R}}^3\) and the geometry of their singular images. Rev. Mat. Iberoam. 31, 33–50 (2015) Oset Sinha, R., Tari, F.: Flat geometry of cuspidal edges. Osaka J. Math. (to appear) (2017). arXiv:1610.08702 Saji, K.: Normal form of swallowtail and its applications (preprint) (2017). arXiv:1703.08904v1 Saji, K., Umehara, M., Yamada, K.: The geometry of fronts. Ann. Math. (2) 169, 491–529 (2009) Takahashi, M.: Envelopes of Legendre curves in the unit tangent bundle over the Euclidean plane. Result Math. 71, 1473–1489 (2017). https://doi.org/10.1007/s00025-016-0619-7 Teramoto, K.: Parallel and dual surfaces of cuspidal edges. Differ. Geom. Appl. 44, 52–62 (2016)