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
Lecture 6 (11.11.2025): Topological spaces, Hausdorff property, Continuity, Topological manifolds, Transition functions, Smooth manifolds (5.1.1, 5.1.2 of MN-GTP), Examples (5.1.3 of MN-GTP), Maps between manifolds and their coordinate representations (5.2.1 of MN-GTP)Â
Lecture 7 (18.11.2025): Ring of smooth functions, Tangent vectors as equivalence classes of curves and derivations of functions, Tangent spaces and tangent bundles, Vector fields (5.2.2 of MN-GTP), Lie bracket (5.3.2 of MN-GTP)
Lecture 8 (25.11.2025): Dual vector spaces, Cotangent spaces and cotangent bundles, 1-forms (5.2.3 of MN-GTP), Tensors, Components of a tensor (5.2.4, 5.2.5 of MN-GTP), Metric (7.1.1 of MN-GTP)