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A parallel manipulator is naturally associated with a set of constraint functions defined by its closure constraints. The differential forms arising from these constraint functions completely characterize the geometric properties of the manipulator. In this paper, using the language of differential forms, we provide a thorough geometric study on the various types of singularities of a parallel manipulator, their relations with the kinematic parameters and the configuration spaces of the manipulator, and the role redundant actuation plays in reshaping the singularities and improving the performance of the manipulator. First, we analyze configuration space singularities by constructing a Morse function on some appropriately defined spaces. By varying key parameters of the manipulator, we obtain homotopic classes of the configuration spaces. This allows us to gain insight on configuration space singularities and understand how to choose design parameters for the manipulator. Second, we define parametrization singularities which include actuator and end-effector singularities (or other equivalent definitions) as their special cases. This definition naturally contains the closure constraints in addition to the coordinates of the actuators and the end-effector and can be used to search a complete set of actuator or end-effector singularities including some singularities that may be missed by the usual kinematics methods. We give an intrinsic classification of parametrization singularities and define their topological orders. While a nondegenerate singularity poses no problems in general, a degenerate singularity can sometimes be a source of danger and should be avoided if possible.