Generic Geometry of Stable Maps of 3-Manifolds into \({\mathbb {R}}^4\)


We describe the generic geometry of the 3D-crosscap (image of a stable map of a 3-manifold into \(\mathbb {R}^4\)) by means of the simultaneous analysis of the generic singularities of height and squared distanced functions on the flag composed by the 3-manifold, the surface of double points and the crosscaps curve at any point of this curve.

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  1. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps. Birkhäuser, Boston (1985)

    Google Scholar 

  2. Arnold, V.I.: Singularity Theory. Selected Papers. LMS Lect. Note Ser. 53. Cambridge University Press, Cambridge (1981)

  3. Bruce, J.W., Kirk, N., du Plessis, A.: Complete transversals and the classification of singularities. Nonlinearity 10(1), 253–275 (1997)

    MathSciNet  Article  Google Scholar 

  4. Bruce, J.W., West, J.: Functions on a crosscap. Math. Proc. Camb. Phil. Soc. 123, 19–39 (1998)

    MathSciNet  Article  Google Scholar 

  5. Casonatto, C., Romero Fuster, M.C., Wik Atique, R.: Topological invariants of stable maps of oriented 3-manifolds in \({\mathbb{R}}^4\). Geom. Dedicata194(1) (2018), 187-207

  6. Golubitsky, M., Guillemin, V.: Stable Mappings and Their Singularities. GTM 14, Springer, New York (1973)

  7. Klingenberg, W.: A Course in Differential Geometry. Springer, New York (1978)

    Google Scholar 

  8. Izumiya, S., Romero Fuster, M. C., Ruas, M. A. S., Tari, F.: Differential Geometry From a Singularity Theory Viewpoint. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2016)

  9. Little, J.:On singularites of submanifolds of higher dimensional Euclidean space. Ann. Mat. Pura Appl. (Ser. 4A)83, 261–336 (1969)

  10. Looijenga, E.J.N.: Structural Stability of Smooth Families of \(C^{\infty }\)—Functions. University of Amsterdam, Thesis (1974)

    Google Scholar 

  11. Mather, J.N.: Stability of \(C^\infty \)-mappings IV: Classification of stable map-germs by \({\mathbb{R}}\)-algebras. Inst. Hautes Études Sci. Publ. Math 37, 223–248 (1970)

    Article  Google Scholar 

  12. Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: The geometry of surfaces in 4-space from a contact viewpoint. Geom. Dedicata54, 323–332 (1995)

  13. Montaldi, J.A.: Contact, with applications to submanifolds of \({\mathbb{R}}^{n}\). PhD Thesis, University of Liverpool (1983)

  14. Monera, M.G., Montesinos Amilibia, A., Sanabria Codesal, E.: The Taylor Expansion of the Exponential Map and Geometric Applications. Rev. R. Acad. Cienc. Exactas F. Nat. Ser. A Math. RACSAM 108 no. 2, 881-906 (2014)

  15. Montaldi, J.A.: On generic composites of maps. Bull. Lond. Math. Soc. 23, 81–85 (1991)

    MathSciNet  Article  Google Scholar 

  16. Nabarro, A.C.: Duality and contact of hypersurfaces in \({\mathbb{R}}^4\) with hyperplanes and lines. Proc. Edinb. Math. Soc. (2) 46(3), 637–648 (2003)

  17. Nabarro, A.C., Romero Fuster, M.C.: \(3\)-Manifolds in Euclidean space from a contact viewpoint. Commun. Anal. Geom.17(4), 755–776 (2009)

  18. Porteous, I.: The normal singularities of a submanifold. J. Differ. Geom. 5, 543–564 (1971)

    MathSciNet  Article  Google Scholar 

  19. Romero Fuster, M.C.: Sphere stratifications and the Gauss map. Proc. R. Soc. Edinburgh95A, 115–136 (1983)

  20. Romero Fuster, M.C.: Semiumbilics and geometrical dynamics on surfaces in 4-spaces. Real Complex Singular. Contemp. Math. Am. Math. Soc. Providence 354, 259–276 (2004)

  21. Wall, C.T.C.: Geometric properties of generic differentiable manifolds. Geometry and topology (Proc. III Latin Amer. School of Math., IMPA, Rio de Janeiro, 1976). Lecture Notes in Math., Vol. 597, Springer, Berlin, pp. 707–774 (1977)

  22. Wik-Atique, R.: On the classification of multi-germs of maps from \(C^2\) to \(C^3\) under \({\cal{A}}\)-equivalence. Real and complex singularities (São Carlos, 1998), 119-133, Chapman and Hall/CRC Res. Notes Math., 412, Chapman and Hall/CRC, Boca Raton, FL (2000)

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M. C. Romero Fuster: MINECO/FEDER (MTM2015-64013-P) R. Wik Atique: Fundação de Amparo à Pesquisa do Estado de São Paulo - FAPESP (2015/04409-6).

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Casonatto, C., Fuster, M.C.R. & Atique, R.W. Generic Geometry of Stable Maps of 3-Manifolds into \({\mathbb {R}}^4\). Bull Braz Math Soc, New Series (2020).

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  • Singularities
  • Flat geometry
  • 3D-crosscap
  • Contacts

Mathematics Subject Classification

  • 57R45
  • 53A07