The original PDF scans often have poor diagrams. Use a modern computational tool (like GeoGebra or Mathematica) to plot the curves and surfaces Struik describes. Struik loves ruled surfaces (hyperboloids, helicoids). See them rotating on your screen.
This article explores the significance of classical differential geometry, why the PDF format has become the standard vessel for its transmission, and how specific lecture notes have shaped the modern understanding of curves, surfaces, and curvature. lectures on classical differential geometry pdf
This theorem shattered the intuition that curvature is purely extrinsic. For example, a cylinder is locally isometric to a plane (one can flatten a cylinder without stretching), and indeed both have (K=0). A sphere ((K>0)) cannot be flattened; a saddle surface ((K<0)) cannot be made planar without distortion. The Theorema Egregium laid the groundwork for Riemannian geometry and eventually Einstein’s general relativity, where gravity is interpreted as intrinsic curvature of spacetime. The original PDF scans often have poor diagrams
where (E = \mathbfx_u \cdot \mathbfx_u), (F = \mathbfx_u \cdot \mathbfx_v), (G = \mathbfx_v \cdot \mathbfx_v). The FFF is the Riemannian metric induced by the ambient Euclidean space. It allows us to compute arc lengths of curves on the surface, angles between tangent vectors, and areas—all without leaving the surface. Two surfaces with the same FFF are said to be ; they are intrinsically identical, even if shaped differently in space (e.g., a plane and a rolled-up sheet of paper). See them rotating on your screen