The Hilbert Projection Theorem underpins projection-based fault detection in linear systems. However, orthogonal projection is inherently linear and cannot be applied directly to nonlinear systems. This chapter addresses projection-based one-class fault detection for nonlinear systems by projecting onto image and residual manifolds, inspired by manifold learning. Using Hamiltonian systems theory, the projection system is constructed by linking the normalised SIR with its adjoint. A major contribution is the use of Bregman divergence as a distance measure in non-Euclidean spaces, integrating control theory with information geometry. The Hamiltonian system projection is interpreted as a geodesic projection onto the image manifold. Concerning uncertainties in data and their estimation, a dual SKR-based Hamiltonian system projecting onto the residual manifold for uncertainty estimation is proposed. This Hamiltonian system is constructed by means of the normalised SKR and its adjoint system. It represents a geodesic projection of the process data onto the residual manifold. This dual SKR-based Hamiltonian system serves as an optimal estimator for such variations in the process data.