Differential Geometry

Differential geometry studies what may be differentiated in form, what survives a change of description, and what local measurement can compel us to conclude about an entire space. This comprehensive progressive treatise begins with analytical, multilinear, curve, and surface foundations, then develops smooth manifolds, tensor fields, metrics, connections, geodesics, curvature, and submanifold geometry. It proceeds to global results involving variation, comparison, completeness, compactness, topology, and characteristic formulae, before treating Lie groups, homogeneous spaces, fibre bundles, gauge geometry, symplectic and complex structures, and the differential geometry of relativity. Six substantial appendices consolidate linear algebra, topology, ordinary differential equations, algebraic topology, coordinate and curvature formulae, and computational differential geometry. Guided examples, colour figures, exercises, and complete resolutions support both sustained study and later reference for readers in mathematics, physics, engineering, computing, medicine, and other quantitative sciences.