Subject: Geometry 4	Code: MMG0005	

Semester	 Hours: C+S+L	Type	 Section	
4	2+1+0	compulsory	Mathematics	

Teaching Staff in Charge	Assoc.Prof. BLAGA Paul Aurel, Ph.D., pablaga@cs.ubbcluj.ro
Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro	
Aims	The course follows in a natural way "Curves and Surfaces" from second semester in the first year by studying the main geometrical objects associated to a differentiable manifolds. The seminars supply by examples,applications, execices and problems the theoretical material given at the course	
Content	1.The space R^n from algebraic and topological point of view. Differentiable mappings.
The locally diffeomorphism theorem and consequences. The rank theorem. Regular points and critical points.
2. Differentiable manifolds. Examples. Topological properties. Differentiable mappings
between manifolds. The tangent space and the tangent map. The cotangent space and the 
cotangent map. Differentiable submanifolds. Immersions, submersions, embedings.
3. Fiber bundles. Vector bundles. Constructions with vector bundles.
4. Vector fields on a manifold. Globally and locally flows. Integrability and completeness. Lie algebra of vector fields.	
Selective References	1. AUSLANDER, L. - MACKENZIE, R.E.: Introduction to Differentiable Manifolds, McGraw-Hill, 1963
2. CONLON, L.: Differentiable Manifolds, Birkhauser, 1993
3. DIECK,  T.:  Topologie, Walter de Gruyter, 1991
4. HIRSCH, M.W.: Differential Topology, Springer, 1976
5. KAHN, D.W.: Introduction to Global Analysis, Academic Press, 1980
6. KOSINSKI, A.: Differential Manifolds, Academic Press, 1993
7. LEE, J.M.: Smooth Manifolds, Springer, 2001
8. MATSUSHIMA, Y.: Differentiable Manifolds, Marcel Dekker, 1972
9. MILNOR, J.W. - Topology from the Differentiable Viewpoint, Princeton University Press, 1997
10. MISHCHENKO, A. - FOMENKO, A.: A course of Differential Geometry and Topology, Mir Publishers, 1988
11. PHAM, F.: Geometrie et calcul differentiel sur les varietes, Dunod, 1999
12. POSTNIKOV, M.M.: Lectures on Geometry, sem. III, Mir Publishers, 1989
13. PRASOLOV, V.V.: Elemente de topologie combinatoriala si diferentiala (în limba rusa), Editura MTNMO, Moscova, 2004	
Assessment	Exam