Classical Differential Geometry I
Course specifics: 6,5 ECTS, 5h x 13 weeks
Schedule for Winter Semester 2023: Tue 3-6pm & Wed 6-8pm
Syllabus
Theory of Curves: Plane curves, curvature, arclength. Curves in R^3. Arclength, curvature, torsion, Frenet frame and formulae. The fundamental theorem (existence and uniqueness). Osculating cycle and sphere. Ruled surfaces, surfaces by revolution, spherical curves.
Theory of surfaces: Curves on a surface. Parameters and reparametrization. Normal and tangent vectors. The tangent plane. The first and the second fundamental form. Gauss, mean, normal and principal curvatures. Christoffel symbols. Gauss map and equations of Gauss and Weingarten. Theorema Egregium. The fundamental theorem of surfaces.
References
A. Arvanitoyeorgos: Elementary Differential Geometry (in Greek)
Th. Vlachos: Differential Geometry of Curves and Surfaces (in Greek)
A. Pressley: Elementary Differential Geometry (Copy in Greek: "Στοιχειώδης Διαφορική Γεωμετρία", Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2011)
B. O'Neill: Elementary Differential Geometry (Copy in Greek: "Στοιχειώδης Διαφορική Γεωμετρία", Ηράκλειο : Πανεπιστημιακές Εκδόσεις Κρήτης, 2002)
J. Oprea: Differential Geometry and its Applications. Prentice Hall, 1997