For a local Lie group M we define cohomology classes [is an obstruction to globalizability and give an example where [w1] = 0. We also show that [w3] coincides with Godbillon-Vey class in a particular case. These classes are secondary as they emerge when curvature vanishes.

This book is about the foundations of geometric symmetry, namely, Lie groups and differential geometry. Although this is a classical subject about which hundreds of books have been written, this book takes a new and innovative approach. The main idea is to replace the Maurer–Cartan form with absolute parallelism and its curvature. Unlike the classical approach, where the model is fixed beforehand by the Maurer–Cartan form, this new approach is model-free, and also revisits the foundational concepts of differential geometry, such as covariant differentiation, from a different perspective.

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This note is an extended version of my presentation at the "Focused Research Workshop on Exterior Differential Systems and Lie Theory" held at the Fields Institute in Toronto, December 9-13, 2013. Some ideas and constructions of this note crystallized during my communication with Anthony D. Blaom. I am deeply grateful to him and also to P.J. Olver for his constant help and encouragement. Prehomogeneous geometriesLet M be a smooth manifold with dim M = n ≥ 2 and j k (f ) p,q be the k-jet of the local diffeomorphism f with source at p and target at q. We call j k (f ) p,q a k-arrow from p to q. Clearly j 0 (f ) p,q = (p, q). Let U p,q k denote the set of all k-arrows from p to q. With the composition and inversion of k-arrows, the set U k def = ∪ p,q∈M U p,q k of all k-arrows on M has the structure of a groupoid

Even though it is quite clear at this stage what curvature means, its technical definition turns out to be problematic, since it needs a technical assumption. However, this assumption forces a projective structure to be flat, and it becomes necessary to modify the approach adopted so far.