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

Lusala, Tsasa

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

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.