The centroaffine Tchebychev operator (Q1895219)

Let \(g\) be the centroaffine metric of a hypersurface immersion \(x : M \to \mathbb{R}^{n + 1}\), \(\nabla\) the corresponding Levi-Civita connection and \(T\) the centroaffine Chebyshev vector field. Then the Chebyshev operator \({\mathfrak I} : TM \to TM\) is defined by \({\mathfrak I} := \nabla T\). \(\mathfrak I\) is selfadjoint with respect to \(g\). The second author showed in a previous paper [Geom. Dedicata 51, No. 1, 63-74 (1994; Zbl 0827.53010)], that the variational problem for the centroaffine surface area leads to \(\text{trace}_g (\nabla T) = 0\). These similarities with the Weingarten operator in affine and Euclidean differential geometry stimulate further discussions on \(\mathfrak I\), carried out in the present paper. One result to be pointed out is the classification of the centroaffine Chebyshev surfaces \(({\mathfrak I} \equiv 0)\).