General Differential Geometries and Related Topics*

space E we mean an m-differentiable manifold H with coordinates in a space T such that (1) T is a space E and H has allowable k coordinate systems^ (2) the differentials of the coordinate transformations possess adjoints with respect to the inner product [x, y] of E at each point of their domains of definition; (3) there exists a covariant vector field valued linear form, called a metric form, g(x, £) in a contravariant vector £ with the following properties at each point x of the coordinate domain of every allowable coordinate system : (3a) g(xf £) is differ entiable in x up to the pth order, where p <m, and where the pth differential is continuous in x; (3b) [fa g(x, £)] is positive definite in £; (3c) g(x, £) is a solvable linear f unction of £ with G(x, rj), say, as its inverse function ; (3d) g(xy £) is self-adjoint (so that [fa, g(x, fa)] is symmetric in fa and fa); (3e) the adjoint g&)(x, fa ôx) of the differential g(x, fa ôx) exists and is itself continuously differ entiable to the (p-l)st order (clearly g**)(x, h àx)=gf3)(xy Ôx; £)); (4) the element of arc length is defined by ds = [ôx, g(x, ôx)]. The theory of a general Riemannian space we call a general Riemannian differential geometry. Define the function T(x, fa, fa) by (23.1) r(*,fa,fa) = G(#, 7 ( * , fa, fa))