Differential invariants in a general differential geometry

Abstract

The study of general differential geometries in which the "geometric space" is a Hausdorff topological space~), while the "coordinate space" i~ a Banach space 3), has been initiated recently by one of us ~). The theory includes as special cases not only the finite dimensional but also the infinite dimensional 5) Riemannian and Non-Riemannian geometries. Partly because of the formidable nature of the calculations in both the finite and infinite dimensional differential invariant theories, special coordinate systems known as normal coordinates 6) play an important role ir~ these theories. In a general differential geometry with Banach coordinates, a coordinate system x (P) is a homeomoiphism mapping a Hausdorff neighborhood of the geometric space H onto an open subset of a fixed open set S of the Banaoh space E, where S is itself the map of some Hausdorff neighborhood by some coordinate system x 0 (P). The value of the function x (P) will be called the coordinate of the point P E H. The intersection of two Hausdorff neighborhoods induces a homeomorphism ~ ~ (x), called a coordinate transformation, taking an open subset of S into an open subset of S.