Differential manifolds 
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Teaching Staff in Charge 
 

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. 
References 
1. AUSLANDER, L.  MACKENZIE, R.E.: Introduction to Differentiable Manifolds, McGrawHill, 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. 