Mathematics BSc
Course description
2013.
Course description
2013.
Differential geometry
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:
| |||
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.