A Note on the Classification of Centroaffine Hypersurfaces

Cao and Wang recently classified a class of locally strongly convex centroaffine hypersurfaces whose non-vanishing Tchebychev vector field $T$ satisfies $K(T,X)=\alpha X$ and $X(\alpha )=0$ for every $X\perp T$, together with $K(T,T)=\beta T$. In the degenerate case $\beta =2\alpha$, their higher-dimensional argument assumes in addition that the $T^{\perp }$-component of the difference tensor $K$ vanishes identically. We show that this extra assumption is unnecessary and the theorem can be strengthened.