Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Graduate
SUBJECT
| MMG0011 | Hyperbolic Geometry |
| Teaching Staff in Charge |
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga cs.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 |
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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 |
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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 |
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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 |