Shared control is a technique to provide interactive autonomy in a telerobotic task, replacing the requirement for pure teleoperation where the operator's intervention is unnecessary or even undesirable. In this thesis, a geometrically correct theory of shared control for teleoperation is developed using differential geometry. The autonomous function proposed is force control. In shared control, the workspace is commonly partitioned into a "position domain" and a "force domain". This computational process requires the use of a metric. In the context of manifolds, these are known as Riemannian metrics. The switching matrix is shown to be equivalent to a filter which embodies a Riemannian metric form. However, since the metric form is non-invariant, it is shown that the metric form must undergo a transformation if the measurement reference frame is moved. If the transformation is not made, then the switching matrix fails to produce correct results in the new measurement frame. Alternatively, the switching matrix can be viewed as a misinterpretation of a projection operator. Again, the projection operator needs to be transformed correctly if the measurement reference frame is moved. Many robot control architectures preclude the implementation of robust force control. However, a compliant device mounted between the robot wrist and the workpiece can be a good alternative in lieu of explicit force control. In this form of shared control, force and displacement are regulated by control of displacement only. The geometry of compliant devices is examined in the context of shared control and a geometrically correct scheme for shared control is derived. This scheme follows naturally from a theoretical analysis of stiffness and potential energy. This thesis unifies some recent results formulated for robotic hybrid position / force control under the modem framework of differential geometry and Lie groups.