Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 1, pp. 275-296 (2002)

Cubic Form Geometry for Surfaces in $ S^3(1)$

Tsasa Lusala

Technische Universität Berlin, Fakultät II: Mathematik und Naturwissenschaften, Institut für Mathematik, Sekr. MA 8-3, Strasse des 17. Juni 136, D-10623 Berlin, Germany, e-mail: lusala@math.TU-Berlin.DE

Abstract: We consider the traceless part $\widetilde C$ of the difference tensor field $C$ between the Levi-Civita connections of the first and the third fundamental forms for non-degenerate surface immersions in $ S^3(1)$. In analogy to affine differential geometry of $ R^{n+1}$ where quadrics are characterized by the vanishing of a traceless cubic form, we study the condition $\widetilde C\equiv0$, give examples and classify non-degenerate surfaces in $ S^3(1)$ which satisfy this condition.

Keywords: nondegenerate surfaces in the $3$-sphere, principal curvature functions, rotational surfaces, cubic form geometry

Classification (MSC2000): 53B25, 53A15, 35-04, 53C24

Full text of the article:

[Previous Article] [Next Article] [Contents of this Number]
© 2002 ELibM for the EMIS Electronic Edition