Abstract
We establish cross-ratio invariants for surfaces in 4-space in an analogous way to Uribe-Vargas’s work for surfaces in 3-space. We study the geometric locii of local and multi-local singularities of orthogonal projections of the surface to 3-space. Cross-ratio invariants at \(P_3(c)\)-points are used to recover two moduli in the 4-jet of a projective parametrization of the surface and identify the stable configurations of the asymptotic curves of the surface.
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References
Banchoff, T., Gaffney, T., McCrory, C.: Cusps of Gauss Mappings Research Notes in Mathematics, p. 55. Pitman (Advanced Publishing Program), London (1982)
Bruce, J.W., Giblin, P.J., Tari, F.: Families of surfaces: height functions, Gauss maps and duals. Pitman Res. Notes Math. 333, 148–178 (1995)
Bruce, J.W., Giblin, P.J., Tari, F.: Families of surfaces: height functions and projections to planes. Math. Scand. 82, 165–185 (1998)
Bruce, J.W., Nogueira, A.C.: Surfaces in \({{{\mathbb{R}}}}^4\) and duality. Q. J. Math. Oxford Ser. Ser. 49(2), 433–443 (1998)
Bruce, J.W., Tari, F.: Families of surfaces in \({{\mathbb{R}}}^4\). Proc. Edinb. Math. Soc. 45, 181–203 (2002)
Bruce, J.W., Tari, F.: Duality and implicit differential equations. Nonlinearity 13, 791–811 (2000)
Cibrario, M.: Sulla reduzione a forma delle equatione linearialle derviate parziale di secondo ordine di tipo misto. Accad. Sci. Lett. Inst. Lomardo Redicconti 65, 889–906 (1932)
Dara, L.: Singularites generiques des equations differentielles multi-forms. Bol. Soc. Bras. Mat. 6, 95–129 (1975)
Davydov, A.A.: Qualitative Control Theory. Translations of Mathematical Monographs, vol. 142. AMS, Providence (1994)
Deolindo-Silva, J.L., Kabata, Y.: Projective classification of jets of surfaces in \(4\)-space. Hiroshima Math. J. 49, 35–46 (2019)
Garcia, R.A., Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: Inflection points and topology of surfaces in \(4\)-space. Trans. Am. Math. Soc. 352, 3029–3043 (2000)
Gibson, C.G.: Singular points of smooth mappings. Pitman Res. Notes Math. 25, 20 (1979)
Gibson, C.G., Hobbs, C.A., Marar, W.L.: On versal unfoldings of singularities for general two-dimensional spatial motions. Acta Appl. Math. 47, 221–242 (1997)
Hobbs, C.A., Kirk, N.P.: On the classification and bifurcation of multi-germs of maps from surfaces to \(3\)-space. Math. Scand. 89, 57–96 (2001)
Izumiya, S., Romero Fuster, M.C., Ruas, M.A.S., Tari, F.: Differential Geometry from Gingularity Theory Viewpoint. World Scientific Publishing Co Pte Ltd, Singapore (2015)
Little, J.A.: On the singularities of submanifolds of heigher dimensional Euclidean space. Annli Mat. Pura et Appl. 4A(83), 261–336 (1969)
Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: The geometry of surfaces in 4-space from a contact viewpoint. Geometr. Dedic. 54, 323–332 (1995)
Mond, D.M.Q.: On the classification of germs of maps from \({{{\mathbb{R}}}}^2\) to \({{{\mathbb{R}}}}^3\). Proc. Lond. Math. Soc. 50, 333–369 (1985)
Nuño-Ballesteros, J.J., Tari, F.: Surfaces in \({\mathbb{R}}^4\) and their projections to 3-spaces. Roy. Proc. Edinburgh Math. Soc. 137A, 1313–1328 (2007)
Oliver, J.: On the characteristic curves on a smooth surface. J. Lond. Math. Soc. 83(3), 755–767 (2011)
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(1), 33–50 (2015)
Platonova, O.A.: Singularities of the mutual disposition of a surface and a line. Uspekhi Mat. Nauk 36(1), 221–222 (1981)
Saji, K.: Criteria for cuspidal \(S_k\) singularities and their applications. J. Gokova Geometry Topol. 4, 67–81 (2010)
Sano, H., Kabata, Y., Deolindo-Silva, J.L., Ohmoto, T.: Classification of jets of surfaces in 3-space via central projection. Bull. Braz. Math. Soc. 48(4), 623–639 (2017)
Uribe-Vargas, R.: A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve. Mosc. Math. J. 6(4), 731–768 (2006)
Wik-Atique, R.: On the Classification of Multi-germs of Maps from \({\mathbb{C}}^2\) to \({\mathbb{C}}^3\) Under \({\cal{A}}\)-Equivalence. Real and Complex Singularities (São Carlos, 1998). Research Notes in Mathematics, vol. 412, pp. 119–133. Chapman & Hall, Boca Raton (2000)
Acknowledgements
The work in this paper is part of the author’s PhD thesis which was supported by the Fundação de Amparo à Pesquisa do Estado de São Paulo, Brazil (FAPESP) Grant no.2012/00066-9. He would like to thank Farid Tari for supervision, Toru Ohmoto for comments and the referee for useful suggestions.
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Deolindo-Silva, J.L. Cross-Ratio Invariants for Surfaces in 4-Space. Bull Braz Math Soc, New Series 52, 591–612 (2021). https://doi.org/10.1007/s00574-020-00221-w
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DOI: https://doi.org/10.1007/s00574-020-00221-w