# Kobayashi's Foundations of Differential Geometry and Pseudogroup of Transformations

I've just started studying Kobayashi's Foundations of Differential geometry and I was wondering why the author introduces the concept of "pseudogroup of transformations" in the very beginning of the book. Up to now I haven't found this concept in anywhere else in mathematics, is it an useful tool? Is that concept really needed for studying differential geometry?

-

A book on differential geometry has to define a smooth manifold somewhere at the beginning. To define a smooth manifold, one has to explain what an atlas is. Kobayashi found a way to shorten the definition of an atlas (and make it easily adaptable to different notions of smoothness) by using the language of a pseudogroup of transformations. The crucial compatibility property of an atlas then simply says that transition maps $\varphi_j\circ \varphi_i$ belong to an appropriate pseudogroup.