## "Babes-Bolyai" University of Cluj-Napoca Faculty of Mathematics and Computer Science

 Riemannian Geometry
 Code Semes-ter Hours: C+S+L Type Section MG004 5 2+1+0 compulsory Matematica MG004 5 2+1+0 optional Matematică-Informatică
 Teaching Staff in Charge
 Prof. VARGA Csaba Gyorgy, Ph.D.,  csvargacs.ubbcluj.roAssoc.Prof. PINTEA Cornel, Ph.D.,  cpinteamath.ubbcluj.ro
 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, Rauch-Riemann comparation theorem, Morse index theorem, the sphere theorem.
 Content 1.Riemannian and pseudoriemannian manifolds. Examples. Euler-Lagrange 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: Hopf-Rinow 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, North-Holland, 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. I-II, 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.: Semi-Riemannian 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.
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject