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

Code	Subject	
MMG0004	Geometry 3	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	3	2+2+0	speciality	compulsory	
Mathematics and Computer Science	3	2+2+0	speciality	compulsory	
Applied Mathematics	3	2+2+0	speciality	compulsory	

Teaching Staff in Charge	Lect. VACARETU Daniel, vacaretu@math.ubbcluj.ro
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro	
Aims	The course gives to the students the principal tools and also the necesary methods in the study of curves and surfaces with an accent on the intuitive side.	
Content	I. Local theory of curves.
1. Curves in euclidean space.Tangent line and normal plane.
2. Curves in general position. Osculating plane.
3. Frenet frame. Frenet formulas. Curvature and torsion.
4. Geometric iterpretation of curvature and torsion.
5. Contact beatween two curves.
6. Evolute and involute.
7. Envelope of a family of curves in plane.
II.Local theory of surfaces.
1. Surfaces in three-dimensional euclidean space.
2. Tangent plane and normal line of a surface.
3. First fundamental form of a surface.
4. Arc length , angle between two curves on surface .Area element.
5. Second fundamental form.
6. Normal curvature.
7. Asymptotic lines of a surface.
8. Principal curvatutres of a surface. Mean curvature and Gaussian curvature.
9. Theorema egregium.
10.Minimal surfaces and surfaces with constant curvature.
11.Darboux frame. Darboux formulas.
12.Geodesic curvature. Geodesic torsion.
13.Geodesic lines.
	
Bibliography	1. BAER, C.: Elementare differentialgeometrie, Walter de Gruyter, 2001
2. DO CARMO, M.P.: Differential Geometry of Curves and Surfaces, Prentice Hall, 1976
3. FEDENKO, A.S.: Culegere de probleme de geometrie diferentiala (în limba rusa), Nauka, Moscova, 1979
4. GRAY, A.:  Modern Differential Geometry of Curves and Surfaces, CRC Press, 1993
5. HSIUNG, C.C.: A First Course in Differential Geometry, John Wiley, 1981
6. KLINGENBERG, W: A Course in Differential Geometry, Springer, 1983
7. KUEHNEL, W.: Differentialgeometrie, Vieweg, 1999
8. MILLMAN, R. - PARKER, G.: Elements of Differential Geometry, Prentice Hall, 1977
9. OPREA, J.: Differential Geometry and its Applications, Prentice Hall, 1997
10. STOKER, J.J.: Differential Geometry, John Wiley, 1969
11. STRUIK, D.J.: Lectures on Classical Differential Geometry, Dover, 1988
12. TAJMANOV, I.A.: Lectii de geometrie diferentiala (în limba rusa), Editura IKI, Moscova, 2002
	
Assessment	30% from the final mark is the activity during one semester
70% from the final mark is the mark from a written test.