The Taylor expansion of the exponential map and geometric applications

The final publication is available at Springer via http://dx.doi.org/10.1007/s13398-013-0149-zIn this work we consider the Taylor expansion of the exponential map of a submanifold immersed in Rn up to order three, in order to introduce the concepts of lateral and frontal deviation. We compute the directions of extreme lateral and frontal deviation for surfaces in R3. Also we compute, by using the Taylor expansion, the directions of high contact with hyperspheres of a surface immersed in R4 and the asymptotic directions of a surface immersed in RnThis work was partially supported by DGCYT grant no. MTM2009-08933.Monera, M.; Montesinos Amilibia, Á.; Sanabria Codesal, E. (2014). The Taylor expansion of the exponential map and geometric applications. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas (RACSAM). 108(2):881-906. https://doi.org/10.1007/s13398-013-0149-zS8819061082Arnol’d, V.I., Gusein-zade, V.I., Varchenko, A.N.: Singularities of Differentiable Maps. Monographs in Mathematics, vol. 82. Birkhäuser, Boston (1985)Chen, B.-Y., Li, S.-J.: The contact number of a Euclidean submanifold. Proc. Edinburgh Math. Soc. 47, 69–100 (2004)Fessler, W.: $$\ddot{U}$$ U ¨ ber die normaltorsion von Fl $$\ddot{a}$$ a ¨ chen im vierdimensionalen euklidischen. Raum. Comm. Math. Helv. 33(2), 89–108 (1959)García, R., Sotomayor, J.: Geometric mean curvature lines on surfaces immersed in $$\mathbb{R}^3$$ R 3 . Annales de la faculté des sciences de Toulouse, $$6^e$$ 6 e ser, vol. 11, No. 3, pp. 377–401 (2002)García, R., Sotomayor, J.: Lines of axial curvature on surfaces immersed in $$R^4$$ R 4 . Differ. Geom. Appl. 12, 253–269 (2000)Golubitsky, M., Gillemin, V.: Stable Mappings and their Singularities. Springer, Berlin (1973)Hartmann, F., Hanzen, R.: Apollonius’s Ellipse and Evolute Revisited–The Discriminant of the Related Quartic. http://www3.villanova.edu/maple/misc/ellipse/Apollonius2004.pdfLlibre, J., Yanquian, Y.: On the dynamics of surface vector fields and homeomofphisms (preprint)Looijenga, E.J.N.: Structural stability of smooth families of $$C^{\infty }$$ C ∞ -functions. University of Amsterdam, Doctoral Thesis (1974)Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: Inflection points and nonsingular embeddings of surfaces in $$\mathbb{R}^5$$ R 5 . Rocky Mt. J. Math. 33, 3 (2003)Mochida, D.K.H., Romero-Fuster, M.C., Ruas, M.A.S.: Osculating hyperplanes and asymptotic directions of codimension two submanifolds of Euclidean spaces. Geom. Dedicata 77(3), 305–315 (1999)Mochida, D.K.H., Romero Fuster, M.C., Ruas, M.A.S.: The geometry of surfaces in 4-space from a contact viewpoint. Geom. Dedicata 54, 323–332 (1995)Monera, G.M., Montesinos-Amilibia, A., Moraes, S.M., Sanabria-Codesal, E.: Critical points of higher order for the normal map of immersions in $$\mathbb{R}^d$$ R d . Topol. Appl. 159, 537–544 (2012)Montaldi, J.A.: Contact with application to submanifolds, PhD Thesis, University of Liverpool (1983)Montaldi, J.A.: On contact between submanifolds. Michigan Math. J. 33, 195–199 (1986)Montesinos-Amilibia, A.: Parametricas4, computer program freely available from http://www.uv.es/montesinMontesinos-Amilibia, A.: Parametricas5, computer program freely available from http://www.uv.es/montesinMoraes, S., Romero-Fuster, M.C., Sánchez-Bringas, F.: Principal configurations and umbilicity of submanifolds in $$I\!\! R^n$$ I R n . Bull. Bel. Math. Soc. 10, 227–245 (2003)Porteous, I.R.: The normal singularities of a submanifold. J. Differ. Geom. 5, 543–564 (1971)Romero-Fuster, M.C., Ruas, M.A.S., Tari, F.: Asymptotic curves on surfaces in $$R^5$$ R 5 . Commun. Contemp. Math. 10, 309–335 (2008)Romero-Fuster, M.C., Sánchez-Bringas, F.: Umbilicity of surfaces with orthogonal asymptotiv lines in $$R^4$$ R 4 . Differ. Geom. Appl. 16, 213–224 (2002)Tari, F.: On pairs of geometric foliations on a cross-cap. Tohoku Math. J. 59(2), 233–258 (2007