Geodesic
HYPERSURFACES IN ${\mathbb C}$P2 AND ${\mathbb C}$H2 WITH TWO DISTINCT PRINCIPAL CURVATURES
Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 137-152

Voir la notice de l'article provenant de la source Cambridge University Press

It is known that hypersurfaces in ${\mathbb C}$Pn or ${\mathbb C}$Hn for which the number g of distinct principal curvatures satisfies g ≤ 2, must belong to a standard list of Hopf hypersurfaces with constant principal curvatures, provided that n ≥ 3. In this paper, we construct a two-parameter family of non-Hopf hypersurfaces in ${\mathbb C}$P2 and ${\mathbb C}$H2 with g=2 and show that every non-Hopf hypersurface with g=2 is locally of this form. 
IVEY, THOMAS A.; RYAN, PATRICK J. HYPERSURFACES IN ${\mathbb C}$P2 AND ${\mathbb C}$H2 WITH TWO DISTINCT PRINCIPAL CURVATURES. Glasgow mathematical journal, Tome 58 (2016) no. 1, pp. 137-152. doi: 10.1017/S0017089515000105
@article{10_1017_S0017089515000105,
     author = {IVEY, THOMAS A. and RYAN, PATRICK J.},
     title = {HYPERSURFACES {IN} ${\mathbb C}${P2} {AND} ${\mathbb C}${H2} {WITH} {TWO} {DISTINCT} {PRINCIPAL} {CURVATURES}},
     journal = {Glasgow mathematical journal},
     pages = {137--152},
     year = {2016},
     volume = {58},
     number = {1},
     doi = {10.1017/S0017089515000105},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000105/}
}
TY  - JOUR
AU  - IVEY, THOMAS A.
AU  - RYAN, PATRICK J.
TI  - HYPERSURFACES IN ${\mathbb C}$P2 AND ${\mathbb C}$H2 WITH TWO DISTINCT PRINCIPAL CURVATURES
JO  - Glasgow mathematical journal
PY  - 2016
SP  - 137
EP  - 152
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000105/
DO  - 10.1017/S0017089515000105
ID  - 10_1017_S0017089515000105
ER  - 
%0 Journal Article
%A IVEY, THOMAS A.
%A RYAN, PATRICK J.
%T HYPERSURFACES IN ${\mathbb C}$P2 AND ${\mathbb C}$H2 WITH TWO DISTINCT PRINCIPAL CURVATURES
%J Glasgow mathematical journal
%D 2016
%P 137-152
%V 58
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089515000105/
%R 10.1017/S0017089515000105
%F 10_1017_S0017089515000105

[1] 1.Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132–141. Google Scholar

[2] 2.Díaz-Ramos, J. C. and Domínguez-Vázquez, M., Non-Hopf real hypersurfaces with constant principal curvatures in complex space forms, Indiana Univ. Math. J. 60 (2011), 859–882. Google Scholar | DOI

[3] 3.Díaz-Ramos, J. C., Domínguez-Vázquez, M. and Vidal-Castiñeira, C., Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, arXiv preprint 1310.0357v1 [math.DG]. Google Scholar

[4] 4.Ivey, T. A., A d'Alembert formula for Hopf hypersurfaces, Results Math. 60 (2011), 293–309. Google Scholar | DOI

[5] 5.Ivey, T. A. and Landsberg, J. M., Cartan for Beginners: Differential geometry via moving frames and exterior differential systems (American Mathematical Society, Providence, RI, 2003). Google Scholar

[6] 6.Kimura, M., Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), 137–149. Google Scholar | DOI

[7] 7.Montiel, S., Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515–535. Google Scholar | DOI

[8] 8.Niebergall, R. and Ryan, P. J., Real hypersurfaces in complex space forms, in Tight and taut submanifolds (Chern, S.-S. and Cecil, T. E., Editors) (MSRI Publications, Cambridge, UK, 1997), 233–305. Google Scholar

[9] 9.Takagi, R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506. Google Scholar

[10] 10.Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43–53. Google Scholar

[11] 11.Warner, F. W., Foundations of differentiable manifolds and Lie groups (Springer-Verlag, New York-Berlin, 1983). Google Scholar | DOI

Cité par Sources :