Tohoku Mathematical Journal

On affine hypersurfaces with parallel second fundamental form

Salvador Gigena

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We investigate the classification problem of hypersurfaces with affine normal parallel second fundamental (cubic) form. A new method of approaching the solution to this problem is here presented; it consists in showing and using the equivalence of the mentioned problem with the classification of a certain class of solutions to the equation of Monge-Ampère type $\det(\pa_{ij}f)=\pm1$.

Article information

Tohoku Math. J. (2), Volume 54, Number 4 (2002), 495-512.

First available in Project Euclid: 11 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A15: Affine differential geometry
Secondary: 35J60: Nonlinear elliptic equations 53C40: Global submanifolds [See also 53B25]


Gigena, Salvador. On affine hypersurfaces with parallel second fundamental form. Tohoku Math. J. (2) 54 (2002), no. 4, 495--512. doi:10.2748/tmj/1113247647.

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  • S. Gigena, General affine geometry of hypersurfaces I, Math. Notae 36 (1992), 1–41.
  • S. Gigena, Constant Affine mean curvature hypersurfaces of decomposable type, Proc. Sympos. Pure Math., Amer. Math. Soc. 54 (1993), 289–316.
  • S. Gigena, Ordinary differential equations in affine geometry: Differential geometric setting and summary of results, Math. Notae 39 (1997/98), 33–59.
  • K. Nomizu and U. Pinkall, Cayley surfaces in affine differential geometry, Tôhoku Math. J. 42 (1990), 101–108.
  • K. Nomizu and T. Sasaki, Affine Differential Geometry, Geometry of affine immersions, Cambridge Tracts in Math. 111, Cambridge University Press, Cambridge, 1994.
  • L. Vrancken, Affine higher order parallel hypersurfaces, Ann. Fac. Sci. Toulouse Math. (5) IX (1988), 341–353.