The numerical calculation of thin beam, plate and shell involves the fourth-order differential equation about deflection whose difficulty lies in constructing approximation functions with C₁ continuity. In the meantime, due to the complexity of the governing equation, the thin curved beam and curved shell are usually simulated approximately by using straight beam or flat plate elements, which is prone to generate geometric errors and then brings errors in mechanical analysis. In our previous study, manifold method based on independent covers is used to analyze curved beam and shell with exact geometry based on the assumption of thick beam and shell. On this basis, the C₁ continuity of the piecewise-defined series solutions of the new manifold method is discussed. The thin curved beam and shell with exact geometry is analyzed based on Euler-Bernoulli beam theory and Kirchhoff-Love shell theory, and the complexity of derivation of geometric formula is overcome. The calculation formula of thin curved beam is given in detail, and the process of thin curved shell is briefly described. The examples in previous study are recalculated under the assumption of thin beam, plate and shell, which verifies the effectiveness of the proposed method. Compared with the assumption of thick beam, plate and shell, the method saves about 30% of the degree of freedom. Meanwhile, the research demonstrates the potential of solving the fourth-order differential equations by applying manifold method based on independent covers.