Mathematics BSc
Course description
2013.

Differential geometry
Hours
lect+pc
Credits
lect+pc
Assessment Specialization Course code
lect/pc
Semester Status
2 + 2 2 + 3 exam +
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
Weak:
practice class
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.