Classical differential geometry measures curves and surfaces in Euclidean three-space by quantities defined at a single point - the curvature and torsion of a unit-speed curve, the principal curvatures and shape operator of a surface - and its characteristic theorems then force global conclusions from pointwise bounds on those quantities. A curve whose curvature stays below a bound must enclose a disc of the reciprocal radius; a complete saddle graph must be a plane; a closed surface's total Gaussian curvature is determined by its topology alone through the Gauss-Bonnet formula; and a bound on Gaussian curvature controls how fat geodesic triangles can be. Recognising which arguments are extrinsic, depending on the embedding, and which survive bending because they depend only on distances measured inside the surface, is the organising distinction, and Gauss's theorema egregium is the statement that curvature falls on the intrinsic side.
Petrunin and Zamora Barrera's "What is differential geometry: curves and surfaces" is a semester-length set of lecture notes, grown out of a 2018 MASS course at Penn State and released under CC BY-SA…