Differential geometry

Mathematics BSc
Course description
2013.

Hours lect+pc	Credits lect+pc	Assessment	Specialization	Course code lect/pc	Semester	Status
2 + 2	2 + 3	exam + term grade	applied math.	mm1c1dg6a mm1c2dg6a	6	optional

Course coordinator

Strong	Weak	Prerequisites
Practice class
		Strong: Geometry1L (mm1c1ge2)
		Strong: Algebra2L (mm1c1al2)
		Strong: Analysis3L-p (mm1c1an3m) or Analysis3L-a (mm1c1an3a)
Lecture
Lecture		Weak: practice class

Remarks

Syllabus designed by:

Prerequisites

Analytical geometry, linear algebra, multivariable calculus.

Literature

Manfredo P. do Carmo: Differential geometry of curves and surfaces. Prentice-Hall, New-Jersey, 1976.
B. Csikós: Differential Geometry. Lecture notes.

Syllabus

Regular smooth curves in the Euclidean space. Length of a curve. Reparameterization and natural reparameterization of a curve. Simple arcs.
Frenet frame of a space curve. Curvature and torsion. Frenet's formulae. Osculating circles. The fundamental theorem of curve theory.
Signed curvature of a planar curve. Evolute, involutes and parallel curves. The rotation number of a closed curve, Umlaufsatz.
Parameterized curves in Rⁿ.
Smooth elementary surface patches in R³. Reparameterization. Linear tangent plane. The unit normal vector field. First fundamental form. Length of a curve on a surface, surface area. Bending of a surface. Smooth surfaces obtained as the preimage of a regular value of a smooth function on R³.
Second fundamental form, normal curvature, Meusnier's theorem. Weingarten map. Principal curvatures and principal directions. Euler's formula. Gauss curvature, Minkowski curvature.
Gauss frame, Christoffel symbols. Stationary and geodesic curves.
Classification of surface points by the sign of the Gauss curvature. Computation of the principal directions. Theorema Egregium (without proof).
Surfaces of revolution. Ruled and developable surfaces. Lines of curvature, Dupin's theorem. Lines of curvature on quadric surfaces. Minimal surfaces.
A glimpse on hypersurfaces in Rⁿ.