Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate

SUBJECT

Code
Subject
MMG0011 Hyperbolic Geometry
Section
Semester
Hours: C+S+L
Category
Type
Mathematics - in Romanian
6
2+1+0
speciality
optional
Applied Mathematics
6
2+1+0
speciality
optional
Teaching Staff in Charge
Prof. VARGA Csaba Gyorgy, Ph.D.,  csvargacs.ubbcluj.ro
Assoc.Prof. PINTEA Cornel, Ph.D.,  cpinteamath.ubbcluj.ro
Aims
The course in an introduction in the Hyperbolic Geometry. It contains elements from the Theory of Surfaces, such as covariant derivative, the geodesics of surfaces, surfaces with constant curvature, the Beltrami model, the Poincare model. We analyze whether the axioms of Hyperbolic Geometry are satisfied.
Content
Courses

Course 1. Normal, principle, total curvature. The Christoffel symbols of a surface
Course 2. The Riemann symbols of a surface. Covariant derivative, parallel transport, the geodetics of surfaces.
Course 3. Revolution surfaces with constant curvature. The geodesics of Revolution surfaces with constant curvature. The Beltrami model.
Course 4. The Poincare model. Curvature and geodetic lines on the Poincare model.
Course 5. The axiomatic system of the absolute Geometry.
Course 6. Proofs of the Incidency axioms. Isometries in the Hyperbolic plane.
Course 7. The hyperbolic distance and its different forms.
Course 8. Congruency cases of the hyperbolic triangles. The sinus and cosines theorems in the hyperbolic triangles.
Course 9. Proofs of the axioms of Archimedes and Cantor in Hyperbolic Geometry. Proofs of the existence of Bolyai-Lobacevski parallels.
Course 10. The parallelism angle of Lobacevski. The area function in the Hyperbolic plane.
Course 11. Geometric locus in the Hyperbolic plane. The Saccheri square.
Course 12. Transformations of Moebius type. The Gauss-Bonet theorem and applications
Course 13. The Fuchs groups and fundamental domains. Constructing the fundamental domains.
Course 14. Modular surfaces and closed geodesics. Arithmetic calculus of geodesic lines. The reduction theorem of Gauss.


SEMINARS

Seminar 1: calculus of Christoffel’s symbols, total curvature of surfaces
Seminar 2: calculus of geodesics of surfaces.
Seminar 3: examples of surfaces with constant curvature
Seminar 4: finding the geodesics of surfaces with constant curvature
Seminar 5: Poincare model
Seminar 6: isometries in the Hyperbolic plane of Poincare
Seminar 7: different forms of the distance in the Hyperbolic plane of Poincare
Seminar 8: applications of the sinus and cosines theorems in solving problems in the Hyperbolic plane
Seminar 9: l Geometric locus in the Hyperbolic plane
Seminar 10: Calculus of the area in the Hyperbolic plane for different geometric figures
Seminar 11: convex sets in the Hyperbolic plane.
Seminar 12: application of the Gauss-Bonet formula
Seminar 13: construction of convex polygons with given angles
Seminar 14: problems with Modular surfaces and closed geodesics
References
1. B.V. Cutuzov, Geometria lui Lobacevski, Editura Tehnică, 1952.
2. D. Brânzei, Geometrie circumstanţială, Editura Junimea Iaşi, 1983.
3. N. V. Efimov, Geomtrie superioară, Editura Tehnică, 1952.
4. S. Katok, Continued fractions, Hyperbolic geometry, Course Notes, 2001.
5. I. Mezei, Cs. Varga, Görbék és felületek elmélete, Egyetemi Jegyzet, Erdélyi Tankönyvtanács, 2002
Assessment
Activity on Seminars 30%
Presentation of a referat 30%
Viva voce final exam 40%
Links: Syllabus for all subjects
Romanian version for this subject
Rtf format for this subject