Abstract
This a survey of the contents of the book with a description of the main topics covered in each chapter. Important classifications of isoparametric, Dupin and Hopf hypersurfaces are described in detail.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)
J. Berndt, Real hypersurfaces in quaternionic space forms. J. Reine Angew. Math. 419, 9–26 (1991)
E. Cartan, Théorie des groupes finis et continus et la géométrie différentielle traitées par la méthode du repère mobile (Gauthiers-Villars, Paris, 1937)
E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante. Annali di Mat. 17, 177–191 (1938)
E. Cartan, Sur quelque familles remarquables d’hypersurfaces (C.R. Congrès Math. Liège, 1939), pp. 30–41
E. Cartan, Sur des familles d’hypersurfaces isoparamétriques des espaces sphériques à 5 et à 9 dimensions. Revista Univ. Tucuman, Serie A 1, 5–22 (1940)
T.E. Cecil, Lie Sphere Geometry, 2nd edn. (Springer, New York, 2008)
T.E. Cecil, P.J. Ryan, Focal sets and real hypersurfaces in complex projective space. Trans. Am. Math. Soc. 269, 481–499 (1982)
T.E. Cecil, P.J. Ryan, Tight and Taut Immersions of Manifolds. Research Notes in Mathematics, vol. 107 (Pitman, London, 1985)
C. Dupin, Applications de géométrie et de méchanique: à la marine, aux points et chausées, etc. (Bachelier, Paris, 1822)
D. Ferus, H. Karcher, H.-F. Münzner, Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177, 479–502 (1981) (see also an English translation by T.E. Cecil (2011) [arXiv:1112.2780v1 [mathDG]])
T.A. Ivey, P.J. Ryan, The structure Jacobi operator for hypersurfaces in CP 2 and CH 2. Results Math. 56, 473–488 (2009)
I.-B. Kim, On Berndt subgroups and extrinsically homogeneous real hypersurfaces in complex hyperbolic spaces, in Proceedings of the Sixth International Workshop on Differential Geometry (Taegu, 2001) (Kyungpook National University, Taegu, 2002), pp. 45–59
M. Kimura, Sectional curvature of holomorphic planes on a real hypersurface in P n(C). Math. Ann. 276, 487–497 (1987)
N. Koike, The quadratic slice theorem and the equiaffine tube theorem for equiaffine Dupin hypersurfaces. Results Math. 47, 69–92 (2005)
S.H. Kon, T.-H. Loo, On Hopf hypersurfaces in a non-flat complex space form with η-recurrent Ricci tensor. Kodai Math. J. 33, 240–250 (2010)
C.R.F. Maunder, Algebraic Topology (Van Nostrand Reinhold, London, 1970)
M. Morse, S. Cairns, Critical Point Theory in Global Analysis and Differential Topology (Academic, New York, 1969)
S. Mullen, Isoparametric systems on symmetric spaces, in Geometry and Topology of Submanifolds VI, ed. by Dillen, F. et al. (World Scientific, River Edge, 1994), pp. 152–154
T. Murphy, Real hypersurfaces of complex and quaternionic hyperbolic spaces. Adv. Geom. 14, 1–10 (2014)
E. Musso, L. Nicolodi, A variational problem for surfaces in Laguerre geometry. Trans. Am. Math. Soc. 348, 4321–4337 (1996)
R. Niebergall, Dupin hypersurfaces in R 5 I, Geom. Dedicata 40, 1–22 (1991)
U. Pinkall, Letter to T.E. Cecil, December 5 (1984)
U. Pinkall, G. Thorbergsson, Deformations of Dupin hypersurfaces. Proc. Am. Math. Soc. 107, 1037–1043 (1989)
U. Pinkall, G. Thorbergsson, Examples of infinite dimensional isoparametric submanifolds. Math. Z. 205, 279–286 (1990)
R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures II. J. Math. Soc. Jpn. 27, 507–516 (1975)
M. Takeuchi, S. Kobayashi, Minimal imbeddings of R-spaces. J. Differ. Geom. 2, 203–215 (1968)
G. Thorbergsson, Tight immersions of highly connected manifolds. Comment. Math. Helv. 61, 102–121 (1986)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Thomas E. Cecil and Patrick J. Ryan
About this chapter
Cite this chapter
Cecil, T.E., Ryan, P.J. (2015). Introduction. In: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3246-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4939-3246-7_1
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-3245-0
Online ISBN: 978-1-4939-3246-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)