Hyperdimensional Spherical Calculus: Advanced Framework for Pure Mathematics Paper III: Differential Geometry and Physical Applications Series: SM-2025-III

This paper presents the third installment in the spherical mathematics series, developing comprehensive tensor calculus and differential geometry on hyperdimensional spherical manifolds. Building upon the algebraic foundations of Paper I and the analytical framework of Paper II, this work establishes advanced geometric structures including covariant derivatives, Riemann curvature tensors, generalized Helmholtz decompositions, and spectral theory of vector Laplacians. The framework provides complete mathematical foundations for analyzing curved spaces with spherical symmetry, with direct applications to Maxwell's equations, Schrödinger equation, and general relativity. All developments include rigorous proofs, computational examples, and numerical implementations, offering both theoretical depth and practical utility for researchers in mathematical physics and differential geometry.