Differential geometry studies the local properties of curves, surfaces, and manifolds using differential and integral calculus, in contrast to topology's treatment of global structure; its historical development traces the transition from Euclidean to non-Euclidean geometry through projective geometry (invariance of geometric properties under projective transformation), the parallel-postulate investigations that led to non-Euclidean geometries, and Riemann's introduction of manifolds and curvature-based geometry, culminating in formalist foundations of geometry. The discipline's core theoretical objects — curvature, geodesics, manifolds, and tensor notation — emerged cumulatively through this historical sequence, with key contributions traceable to figures including Monge, Poncelet, Gauss, Bolyai, Lobachevsky, Riemann, Beltrami, Klein, Poincaré, and Hilbert, and the field underlies later applications in general relativity. This is a topic in the history and philosophy of mathematics as applied to differential and non-Euclidean geometry, situating differential geometry's technical content within its intellectual and disciplinary lineage.
Differential geometry studies the local properties of curves, surfaces, and manifolds using differential and integral calculus, in contrast to topology's treatment of global structure; its historical…