The sequence of nineteenth-century steps that turned geometry from the study of figures in Euclidean space into the study of abstract spaces. Gauss's 1828 Disquisitiones defines the Gauss map of a surface in three-space, computes its curvature as the Jacobian determinant of that map, and proves the Theorema Egregium: the curvature depends only on the first fundamental form, so it is intrinsic to the surface. In parallel the projective school of Monge, Carnot, Poncelet and Chasles builds a synthetic geometry organised by the invariance of the cross-ratio and by duality between points and lines, which Mobius, Plucker and Grassmann then made algebraic through barycentric and homogeneous coordinates, line coordinates and the exterior product, and which Klein's Erlangen Program recast as the study of the invariants of a transformation group. Riemann's 1854 Habilitationsvortrag joins the two by positing an n-dimensional manifold with no ambient space at all, carrying a positive-definite quadratic differential form ds^2 = sum g_ij dx_i dx_j; expanding the metric in geodesic normal coordinates produces a biquadratic correction term whose coefficients are the Riemann curvature tensor, generalising Gauss's intrinsic curvature to any dimension. The closing thread is the machinery needed to make 'space' precise at all -- Gauss's geometric reading of complex numbers, Cantor's set theory, the metric and Hausdorff space definitions of Frechet and Hausdorff, Poincare's Analysis Situs, and Weyl's 1913 axiomatisation of a Riemann surface as the first abstract manifold.
A historical essay by Raymond O. Wells, Jr. on three developments in 19th-century geometry that together produced the modern notion of a manifold. First, Gauss's 1828 Disquisitiones generales circa s…