Abstract
In this paper we write some differential formulas involving the high-order Levi curvatures of a real hypersurface in a complex space form. As application we get a classification result under suitable assumptions.
Similar content being viewed by others
References
Berndt, J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)
Bogges, A.: CR Manifolds and the Tangential Cauchy–Riemann Complex. Studies in Advanced Mathematics (1991)
Borisenko, A.A.: On the global structure of Hopf hypersurfaces in a complex space form. Ill. J. Math. 45, 265–277 (2001)
Dileo, G., Lotta, A.: Levi-parallel contact Riemannian manifolds. Math. Z. 274, 701–717 (2013)
Klingenberg, W.: Real hypersurfaces in Kähler manifolds. Asian J. Math. 5, 1–17 (2001)
Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Am. Math. Soc. 296, 137–149 (1986)
Kon, M.: On a Hopf hypersurface of a complex space form. Differ. Geom. Appl. 28, 295–300 (2010)
Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley, London (1969)
Hounie, J., Lanconelli, E.: An Alexandrov type theorem for Reinhardt domains of \({\mathbb{C}}^2\). In: Recent progress on some problems in several complex variables and partial differential equations. Contemp. Math. 400, 129–146 (2006)
Lanconelli, E., Montanari, A.: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differ. Equ. 202, 306–331 (2004)
Maalaoui, A., Martino, V.: A symmetry result on submanifolds of space forms and applications. Mediterr. J. Math. 10, 507–517 (2013)
Martino, V.: A symmetry result on Reinhardt domains. Differ. Integral Equ. 24, 495–504 (2011)
Martino, V., Montanari, A.: On the characteristic direction of real hypersurfaces in \(\mathbb{C}^{N+1}\) and a symmetry result. Adv. Geom. 10, 371–377 (2010)
Martino, V., Montanari, A.: Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems. Forum Math. 22, 255–267 (2010)
Martins, J.K.: Hopf hypersurfaces in space forms. Bull. Braz. Math. Soc. (N.S.) 35, 453–472 (2004)
Miquel, V.: Compact Hopf hypersurfaces of constant mean curvature in complex space forms. Ann. Global Anal. Geom. 12, 211–218 (1994)
Monti, R., Morbidelli, D.: Levi umbilical surfaces in complex space. J. Reine Angew. Math. 603, 113–131 (2007)
Montiel, S.: Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 37, 515–535 (1985)
Montiel, S., Romero, A.: On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20, 245–261 (1986)
Niebergall, R., Ryan, P.J.: (1997) Real hypersurfaces in complex space forms. In: Tight and Taut Submanifolds (Berkeley, CA, 1994), vol. 32, pp. 233–305. Mathematical Sciences Research Institute Publication, Cambridge University Press, Cambridge
Okumura, M.: On some real hypersurfaces of a complex projective space. Trans. Am. Math. Soc. 212, 355–364 (1975)
Ryan, P.J.: Hypersurfaces with parallel Ricci tensor. Osaka J. Math. 8, 251259 (1971)
Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)
Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures. J. Math. Soc. Japan 27, 43–53 (1975)
Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures. II. J. Math. Soc. Japan 27, 507–516 (1975)
Tralli, G.: Levi curvature with radial symmetry: a sphere theorem for bounded Reinhardt domains of \(\mathbb{C}^2\). Rend. Sem. Mat. Univ. Padova 124, 185–196 (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Martino, V., Tralli, G. High-order Levi curvatures and classification results. Ann Glob Anal Geom 46, 351–359 (2014). https://doi.org/10.1007/s10455-014-9427-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-014-9427-z