Closed Surface Integral Geometry A Tensorial Moment Formulation

This paper develops a rigorous integral–tensor formulation of closed hypersurface geometry in Rn\mathbb{R}^nRn. Departing from classical curvature-based differential geometry, the study establishes a global characterization of compact, oriented C2C^2C2 hypersurfaces without boundary through a hierarchy of normal moment tensors defined by surface integrals of tensor powers of the unit normal field. The fundamental first-order identity, derived from the divergence theorem, provides a necessary closure condition. Higher-order symmetric normal tensors are introduced systematically, forming an infinite tensor hierarchy that encodes geometric structure. Parity arguments establish the vanishing of odd-order tensors under central symmetry, while isotropic tensor classification proves that for spheres of class C∞C^\inftyC∞, all even-order tensors reduce to canonical combinations of Kronecker deltas. The framework further incorporates mixed position–normal tensors and classical divergence identities, including volume representation formulas. A reconstruction conjecture is proposed: under appropriate smoothness assumptions, a closed hypersurface is uniquely determined, up to rigid motion, by its complete infinite normal tensor spectrum. This conjecture connects the theory to multidimensional moment problems and convex geometric analysis. The resulting formulation provides a coordinate-free, moment-theoretic foundation for hypersurface geometry in arbitrary dimension, complementing traditional local curvature approaches with a global tensor-invariant perspective.