Riemannian geometry 
ter 

Teaching Staff in Charge 
 

Aims 
The main purpose of the course consists in construction of the principal instruments which are necessary in studying the Riemann geometry. The following notions and results are studied: Jacobi fields, isometric inversions, constant curvature spaces, the variation of the energy integral, RauchRiemann comparation theorem, Morse index theorem, the sphere theorem. 
Content 
1.Riemannian and pseudoriemannian manifolds. Examples. EulerLagrange equations of some
integral type. Geodesics. Riemannian connexion. The tensor of Riemann and Riemannian curvature. 2.Jacobi'fields: Jacobi' equation, conjugate points. The second fundamental form. Fundamental equation. 3.Complete Riemannian manifolds: HopfRinow theorem. Hadamard'theorem. Hyperbolic sapces. The isometries of the hyperbolic spaces. The first and the second variation formula of the energy integral. Rauch' comparasion theorem and applications. Morse' index formula. The sphere' theorem. 
References 
1. BESSE, A.E.: Einstein Manifolds, Springer, 1987
2. BOOTHBY, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry (ed. a doua), Academic Press, 1986 3. DO CARMO, M.P.: Riemannian Geometry, Birkhauser, 1992 4. CHAVEL, I.: Riemannian Geometry: A Modern Introduction, Cambridge University Press, 1993 5. CHEEGER, J.  EBIN, D.G.: Comparison Theorems in Riemannian Geometry, NorthHolland, 1975 6. CHERN, S.S.  CHEN, W.H.  LAM, K.S.: Lectures on Differential Geometry, World Scientific, 1999 7. GALLOT, S.  HULIN, D.  LAFONTAINE, J.: Riemannian Geometry, Springer, 1987 8. GOLDBERG, S.I.: Curvature and Homology, Dover, 1998 9. KOBAYASHI, S.  NOMIZU, K.: Foundations of Differential Geometry, vol. III, Interscience, 1963, 1969 10. LEE, J.M.: Riemannian Manifolds: An Introduction to Curvature, Springer, 1997 11. MORGAN, F.: Riemannian Geometry: A Beginner's Guide, Jones and Bartlett, 1993 12. O'NEILL, B.: SemiRiemannian Geometry with Applications to General Relativity, Academic Press, 1983 13. POSTNIKOV, M.M.: Geometry VI: Riemannian Geometry, Springer, 2001 
Assessment 
30% from the final mark is the activity during one semester
70% from the final mark is the mark from a written test. 