Tchebychev surfaces of \(\mathbb{S}^3(1)\) with constant curvature functions
A hypersurface in a space form is called nondegenerate if its Weingarten operator has maximal rank. Denote by \(\nabla\)(I), \(\nabla\)(II), \(\nabla\)(III) the Levi-Civita connections associated to the three fundamental forms I, II, III, resp., and by \(2C\) the symmetric \((1,2)\)-tensor \(\nabla\)(I)--\(\nabla\)(III). Its traceless part \(\widetilde C\) is an interesting geometric object, it has properties analogous to that of the cubic form in affine hypersurface theory. The author introduces a new class of interesting hypersurfaces, called Tchebychev hypersurfaces, characterized by the property that the covariant derivative \(\nabla\)(II)\(\widetilde C\) is totally symmetric. The author classifies Tchebychev surfaces in the 3-sphere \(\mathbb{S}^3(1)\) with particular curvature properties.
scientific article; zbMATH DE number 1498583