This paper analyzes the geometric quantities that remain unchanged during parallel mapping (i.e., mapping from a reference curved surface to a parallel surface with identical normal direction). The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invariants or geometrically conserved quantities. These include not only local mapping invariants but also global mapping invariants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invariants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invariants and transformations have potential applications in geometry, physics, biomechanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.

Based on the second gradient operator and corresponding integral theorems such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem on curved surfaces, a few new scalar differential operators are defined and a series of integral transformations are derived. Interesting transformations between the average curvature and the Gauss curvature are presented. Various conserved integrals related to the Gauss curvature and the second fundamental tensor are disclosed. The important applications of the results in disciplines such as the geometry, physics, mechanics, and biology are briefly discussed.

A new gradient operator was derived in recent studies of topological structures and shape transitions in biomembranes. Because this operator has widespread potential uses in mechanics, physics, and biology, the operator’s general mathematical characteristics should be investigated. This paper explores the integral characteristics of the operator. The second divergence and the differential properties of the operator are used to demonstrate new integral transformations for vector and scalar fields on curved surfaces, such as the second divergence theorem, the second gradient theorem, the second curl theorem, and the second circulation theorem. These new theorems provide a mathematical basis for the use of this operator in many disciplines.