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
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