Section numbers labelled AP-EDG are from the textbook Elementary Differential Geometry by Andrew Presley.
Section numbers labelled AM-FSDG are from the textbook First Steps in Differential Geometry by Andrew McInerney.
Lecture 1 (25.02.2025): Introduction, Brief history of geometry, Curves, Level curves, Cartesian equation, Parametrized curves, Parameter and trace, Non-uniqueness of parametrization, Plane and space curves, Tangent vectors, Simple, smooth and regular curves (Sec 1.1 of AP-EDG)
Lecture 2 (04.03.2025): Arclength, Speed of a curve, Unit-speed curves, Reparametrization, Curves parametrized by arclength, Unit tangent vector, Curvature vector and curvature, Unit normal vector, Unit binormal vector for space curves (Sec 1.2, 1.3, 2.1 of AP-EDG)
Lecture 3 (11.03.2025): Unit tangent, unit normal and unit binormal vectors for space curves, Curvature vector and curvature, Radius of curvature, Frene-Serret frame, Torsion, Frenet-Serret equations (Sec 2.1, 2.3 of AP-EDG)
Lecture 4 (18.03.2025): Frenet-Serret frame and equations for arbitrary parametrization, Curvature and torsion formulas for space curves, Geometric intuition behind curvature and torsion, Fundamental theorem of space curves (Sec 2.3 of AP-EDG)
Lecture 5 (25.03.2025): Preliminaries: open balls, open sets, continuity, homeomorphisms, smooth maps, diffeomorphisms, Coordinate charts or surface patches, Surfaces, Regularity, Examples: plane, cylinder, sphere, Reparametrization, Tangent plane, Unit normal vector (Sec 4.1, 4.2, 4.4, 4.5 of AP-EDG)
Lecture 6 (15.04.2025): Transition maps, Smooth maps between surfaces, Derivatives of such maps, Jacobian matrix, Orientability, First fundamental form, Length of a curve on a surface (Sec 4.3, 6.1 of AP-EDG)
Lecture 7 (22.04.2025): Local isometries, Isometries, Area of a region on a surface, Recall of the notion of curvature for curves via Taylor expansion, Extension of this notion to surfaces, Second fundamental form (Sec 6.2, 6.4, 7.1 of AP-EDG)
Lecture 8 (29.04.2025): Orientable and oriented surfaces, Gauss and Weingarten map, Gaussian and mean curvatures, Principle curvatures and corresponding principle vectors, Christoffel symbols, Codazzi-Mainardi and Gauss equations, Gauss' theorema egregium (Sec 7.2, 8.1, 8.2, 10.1, 10.2 of AP-EDG and Prop 7.4.4)
Lecture 9 (06.05.2025): Topological spaces, Hausdorff spaces, Continuous maps and homeomorphisms, Local charts, Topological manifolds, Transition maps, Smooth manifolds, Smooth maps between smooth manifolds and diffeomorphisms, Ring of real valued smooth functions, Tangent vectors as equivalence class of curves, Tangent spaces and their vector space structure (spanned by partial derivatives), Tangent bundles and their manifold structures, Derivations over a ring, Vector fields as derivations over the ring of real valued smooth functions, Cotangent spaces as duals of tangent spaces, 1-forms, Cotangent bundles, Tensors (Sec 3.2, 3.4, 3.7 of AM-FSDG) Read also: Sec 2.3, 2.5, 2.6, 2.9 of AM-FSDG.
Lecture 10 & 11 (13.05.2025 - 20.05.2025): Tangent and cotangent bundles, Local frames and coframes, Holonomic and non-holonomic local frames, Lie bracket of vector fields, p-forms, Wedge product, Exterior derivative, Lie derivative, Interior product, Cartan calculus relations, Metric tensor, Hodge star isomorphism, Reconstruction of 3D calculus operators (gradient, Laplacian, divergence, curl, scalar product, vector product) in terms of exterior derivative, Hodge star and wedge product, Examples in R^3 with Cartesian and spherical coordinates (Sec 4.1, 4.2, 4.3, 4.4, 4.6, 4.7; See also lecture notes provided in the announcements page)
Lecture 12 (27.05.2025): Metric and inverse metric, Affine connections as generalizations of directional derivatives, Parallelism, Metric-affine geometry, Torsion, Curvature, Non-metricity, Levi-Civita connection, Fundamental theorem of Riemannian geometry, Schouten decomposition, Ricci tensor, Ricci scalar, Einstein tensor, Einstein field equations, Closing remarks (Sec 5.1, 5.3, 5.4, 5.5; See also lecture notes provided in the announcements page)