The geometry of G-structures

CHERN 1. Introduction. Differential geometry studies differentiate manifolds and geometric objects or structures on them. It is now customary to distinguish it from differential topology by the presence of a structure in addition to the differentiable structure. What a differential geometric structure is or should be is a matter of taste. At the present state of the field a definition general enough to include all the significant structures will certainly contain many uninteresting ones. Among the attempts at a general definition is the notion of a geometric object initiated by Oswald Veblen [l30].