Atlas of Fundamental Concepts

Local Euclidean charts turn topological spaces into manifolds; smooth compatibility enables calculus. Tangent and cotangent bundles support vector fields, tensors, and differential forms. Connections differentiate geometric data, curvature measures noncommutativity of covariant differentiation, and geodesics encode distinguished motion. Metrics, symplectic forms, Poisson brackets, complex structures, Hermitian metrics, Kähler compatibility, contact forms, and Lorentzian signature produce major geometric families.