The Foundations of Differential Geometry

a₁ affine connection affine geometry affine transformation allowable coordinate system arithmetic n-cell arithmetic point arithmetic space arithmetic straight line axioms G called cartesian coordinate system centred affine space components correspondence Cramer's rule curve defined determines differential equations differential geometry dimensions dx³ equivalent Erlanger Programm Euclidean geometry Euclidean metric Euclidean space flat affine space flat k-space follows function geometric object given point holonomic inverse linear homogeneous transformation linear k-space linear transformation locally flat affine manifold of class Math matrix metric spaces n-cell containing n-cell of class non-singular ordinary point oriented manifolds preferred coordinate system Q. F. Chap region regular manifold regular point transformations regular transformation represented Riemannian spaces satisfying the axioms scalars set of points simple manifold space satisfying structure tangent space tensors theorem theory transformation group transformation of class transformation of coordinates transformation which carries translation vector