Section numbers labelled AP-EDG are from the textbook Elementary Differential Geometry by Andrew Presley.
Section numbers labelled MN-GTP are from the textbook Geometry, Topology and Physics by Mikio Nakahara.
Lecture 1 (23.09.2025): Introduction to the course, Syllabus
Lecture 2 (30.09.2025): A little history of geometry, Curves (1.1 of AP-EDG), Parametrized curves, Tangent vectors, Simple and regular curves, Archlength (1.2 of AP-EDG), Speed, Reparametrization (1.3 of AP-EDG), Archlength paramentrization
Lecture 3 (07.10.2025): Space curves, Unit tangent, unit normal, unit binormal, curvature vectors, Curvature (2.1 of AP-EDG), Radius of curvature, Torsion, Frenet-Serret equations (2.3 of AP-EDG) , Frenet-Serret frame, Fundamental theorem for space curves
Lecture 4 (14.10.2025): Coordinate charts or surface patches, Surfaces (4.1 of AP-EDG), Smoothness and regularity (4.2 of AP-EDG), Tangent plane (4.4 of AP-EDG), Unit normal vector (4.5 of AP-EDG), First fundamental form (6.1 of AP-EDG), Length of a curve on a surface, Area of a region on a surface (6.2 of AP-EDG)
Lecture 5 (21.10.2025): Smooth maps between surfaces (4.3 of AP-EDG), Derivatives of smooth maps and the Jacobian matrix, Transition maps, Second fundamental form (7.1 of AP-EDG), Orientability, Gauss and Weingarten maps (7.2 of AP-EDG), Gaussian and mean curvatures (8.1 of AP-EDG), Principle curvatures and corresponding principle vectors (8.2 of AP-EDG), Gauss' theorema egregium