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Voltage instability is one of the main causes of power system blackouts. Emerging technologies such as renewable energy integration, distributed energy resources and demand responses may introduce significant uncertainties in analyzing of system-wide voltage stability. This paper starts with summarizing different known voltage instability mechanisms, and then focuses on a class of voltage instability which is induced by the singular surface of the algebraic manifold. We argue and demonstrate that this class can include both dynamic and static voltage instabilities. To determine the minimum distance to the point of voltage collapse, a new formulation is proposed on the algebraic manifold. This formulation is further converted into an optimal control framework for identifying the path with minimum distance on the manifold. Comprehensive numerical studies are conducted on some manifolds of different power system test cases and demonstrate that the proposed method yields candidates for the local shortest paths to the singular surface on the manifold for both the dynamic model and the static model. Simulations show that the proposed method can identify shorter paths on the manifold than the paths associated with the minimum Euclidean distances. Furthermore, the proposed method always locates the right path ending at the correct singular surface which is responsible for the voltage instability; while the Euclidean distance formulation can mistakenly find solutions on the wrong singular surface. A broad range of potential applications using the proposed method are also discussed.