On T H E Foundations of General Infinitesimal Geometry* by Hermann Weyl

In connection with a seminar on infinitesimal geometry in Princeton, in which I took part, it seemed desirable to clarify the relations between the work of the Princeton school and that of Cartan. With a group © of transformations in m variables £ is associated, in accordance with Klein's Erlanger Program, a homogeneous or plane space 9? of the kind © ; a point of 9? is represented by a set of values of the "coordinates" £ and figures which go into each other on subjecting the coordinates to a transformation of © are to be considered as fully equivalent. The transformations of © give at the same time the transition between two allowable "normal" coordinate systems in 9?. If we have two spaces 9î, 9?' of the kind © and set up a definite normal coordinate system in each of them, then such a transformation can be interpreted as an isomorphic representation of 9? on 9Î'. © is assumed to be transitive. Cartan f developed a general scheme of infinitesimal geometry in which Klein's notions were applied to the tangent plane and not to the n-dimensional manifold M itself. The