Centroaffine differential geometry and its relations to horizontal submanifolds

We relate centroaffine immersions $f:M^n\to R^{n+1}$ to horizontal immersions $g$ of $M^n$ into $S^{2n+1}_{n+1}(1)$ or $H^{2n+1}_n(-1)$. We also show that $f$ is an equiaffine sphere, i.e. the centroaffine normal is a constant multiple of the Blaschke normal, if and only if $g$ is minimal