In this paper, a mixed tensor analysis for a two-dimensional (2D) manifold embedded into a three-dimensional (3D) Riemannian space is conducted and its applications to construct a dimensional splitting method for linear and nonlinear 3D elastic shells are provided. We establish a semi-geodesic coordinate system based on this 2D manifold, providing the relations between metrics tensors, Christoffel symbols, covariant derivatives and differential operators on the 2D manifold and 3D space, and establish the Gateaux derivatives of metric tensor, curvature tensor and normal vector and so on, with respect to the surface $\vec{\Theta}$ along any direction $\vec{\eta}$ when the deformation of the surface occurs. Under the assumption that the solution of 3D elastic equations can be expressed in a Taylor expansion with respect to transverse variable, the boundary value problems satisfied by the coefficients of the Taylor expansion are given.