KM Space Theory, Volume III: Relative Metric Connections, Curvature, and Differential Geometry

We develop the differential-geometric and cohomological layer of KM space theory through typed comparisons between admissible differential realizations. For two metrics \(g_0,g_1\) and two connections \(\nabla^0,\nabla^1\), we study the discrepancy operators \[ B=g_0^{-1}g_1, \qquad A=\nabla^1-\nabla^0, \] together with the curvature identity \[ R^1-R^0=d_{\nabla^0}A+A\wedge A. \] We construct relative torsion, curvature, parallel transport, holonomy, Jacobi fields, Hodge operators, Green operators, characteristic forms, transgression classes, differential characters, and Gauss–Manin connections under explicitly stated regularity, constant-rank, compactness, algebraicity, and comparison hypotheses. We distinguish vector bundles from rank-jumping pseudobundles, ordinary cohomology from reduced Hilbert-complex cohomology, canonical decompositions from noncanonical splittings, and differential characteristic classes from algebraic cycles. Constant-rank Moore–Penrose calculus, spectral stratification, quotient connections, prescribed-torsion metric connections, Hodge decomposition, Chern–Weil and Chern–Simons identities, Smith-normal-form homology, universal-coefficient sequences, torsion linking pairings, and Hopkins–Singer-type differential cohomology are developed with their naturality and base-change laws. A homotopy-coherent multiplicative datum yields a natural unital associative graded-commutative product \[ \widehat H_{\mathrm{KM}}^p \times \widehat H_{\mathrm{KM}}^q \longrightarrow \widehat H_{\mathrm{KM}}^{p+q}. \] Relative products and fiber integration are characterized by an explicit coherent transfer package. Algebraic connections, connection-type \(\mathcal D\)-modules, Picard–Vessiot–Tannakian comparison, smooth-proper Gauss–Manin theory, derived degeneration, Deligne realization, and reverse comparison are treated through typed obstruction and lifting theories. Comparison defects are organized by a natural seven-term kernel–cokernel exact sequence, while every reverse lifting problem is expressed by a cokernel obstruction and a kernel torsor. Counterexamples delimit all conclusions that fail without the declared hypotheses. **Keywords** KM space theory; relative metric; relative connection; curvature; torsion; holonomy; Hodge theory; Green operator; Chern–Weil theory; Chern–Simons transgression; differential cohomology; Hopkins–Singer complex; Smith normal form; torsion linking pairing; Gauss–Manin connection; differential Galois theory; \(\mathcal D\)-modules; Deligne cohomology; rank-jumping pseudobundle; comparison defect; obstruction theory.