For an \(n \times n\) matrix whose minimal polynomial has degree n, a new class of curves in \({\textbf{R}}^n\) is introduced, which we call reduced centroaffine curves. We show that a reduced centroaffine curve lies on an explicitly described centroaffine homogeneous surface if the curvature satisfies a certain simple condition. We meet such curves in our previous work on centroaffine surfaces of cohomogeneity one as their profile curves. Since these curves do not seem pure inhabitants of centroaffine geometry, we introduce a new corresponding concept of curvature. This curvature is the \((n-1)\) -dimensional unit sphere valued function. As a first step, we forcus on reduced centroaffine curves whose curvature lies on a great hypersphere. In the case where n is equal to 2, such a condition means that the curvature is constant. In the appendix, we provide a classification table of centroaffine homogeneous surfaces, including the degenerate ones that appear in the theorem above.
