Preprint 676-2000

Cubic form geometry for surfaces in S^3(1)

Author(s) : Tsasa Lusala

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 676-2000

MSC 2000

53B25 Local submanifolds

53A15 Affine differential geometry

35-04 Explicit machine computation and programs

53C24 Rigidity results

Abstract :
We consider the traceless part $\widehat{C}$ of the difference tensor field $C$ between the Levi-Civita connections of the first and the second fundamental forms for non-degenerate surface immersions in $\S^3(1)$. In analogy to affine differential geometry of $\mathbb{R}^{n+1}$ where quadrics are characterized by the vanishing of the traceless cubic form, we study the condition $\widehat{C}=0$, give examples and classify non-degenerate surfaces in $S^3(1)$ which satisfy this condition.

Keywords : Non-degenerate surfaces in 3-spheres, principal curvature functions, rotational surfaces, cubic form geometry