Using basic tools of Euclidian space differential geometry, maximal singularity free components of the regular manipulator configuration space are defined, with conditions that establish the space as a differentiable manifold. This structure shows that the conventional categorization of manipulators as either serial or parallel is incomplete and that three distinct categories of manipulator must be accounted for; (1) serial manipulators in which inputs globally determine outputs, (2) explicit parallel manipulators in which outputs globally determine inputs, and (3) compound manipulators in which there is no global input or output mapping. Results of differential geometry are used to show that configuration space differentiable manifolds in each category are partitioned into maximal, disjoint, path-connected components in which the manipulator is singularity free and may be effectively controlled. This extends local analytical properties of manipulators that are used for analysis and control to global validity on maximal components of regular manipulator configuration space, providing explicit criteria for avoidance of singular behavior. Model manipulators in each of the three categories are analyzed to illustrate application of the differentiable manifold structure, using only multivariable calculus and linear algebra. Computational methods for forward and inverse kinematics and construction of ordinary differential equations of manipulator dynamics on differentiable manifolds are presented in Part II of this paper, in support of manipulator control.