Gauss, Normals and Fundamental Forms in Differential Geometry - Lecture 34

Explore Gauss's approach to differential geometry in this comprehensive lecture. Delve into parametric descriptions of surfaces and the Gauss-Rodrigues map, which illustrates how unit normals move along a surface. Learn about the first fundamental form, describing Euclidean quadratic forms in terms of parametrization, and the second fundamental form, determined by the Gauss-Rodrigues map's derivative. Examine examples involving spheres and other surfaces, and investigate the classification of surface points. Discover Gauss's groundbreaking Theorema Egregium from 1827, which revolutionized the field of differential geometry.

Introduction
C.F.Gauss1777-1855
1st fundamental formI.e quadratic form
Gauss introduced the idea of a surface S parametrically
Gauss- Rosrigues map
Gauss realised that the Gaussian curvature can be obtained by
Ex.1 Sphere radius
Ex.2
Ex.3
Interesting questions- differentiating points on a surface S into
Parabolic points
Theorema Egregiurn 1827