Geometric Homogeneity and Stabilization

We present a consistent, geometric notion of homogeneity, for vector elds (diierential equations and control systems), functions, diierential forms and endomorphisms. The fundamental observation is that homogeneity is an intrinsic, geometric property. Accordingly, a coordinate-free characterization is given, and shown to yield the desired/expected results. This also addresses: \When do there exist local coordinates such that in these coordinates a vector eld is homogeneous in the traditional sense?" While no simple algebraic test is known, the coordinate-free approach still allows one to exploit the geometric homogeneity properties without having to transform the objects into homogenizing local coordinates. We clarify which objects to declare homogeneous of degree zero, tracing the persisting confusion to a common mix-up of vector elds and endomorphisms. 1. WHY HOMOGENEITY? Aside from aesthetic aspects, mathematical objects that exhibit homogeneity properties often allow for substantial simpliications, not infrequently even closed-form analytic solutions. The most simple case is when the objects are linear; the key to projective geometry is homogeneity or homogenization; scalar diierential equations that are homogeneous become separable after a simple change of variable, and so on.