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

Riemannian geometry
Code
Semes-
ter
Hours: C+S+L
Credits
Type
Section
MG004
5
2+1+0
6
compulsory
Matematică
MG004
5
2+1+0
5
optional
Matematică-Informatică
Teaching Staff in Charge
Assoc.Prof. VARGA Csaba Gyorgy, Ph.D., csvarga@cs.ubbcluj.ro
Assoc.Prof. PINTEA Cornel, Ph.D., cpintea@math.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. Carmo, M. do, Riemannian Geometry, Birkhausel, Boston, Berlin, 1992.
2. Gh. Gheorghiev, V. Oproiu, Varietati finit si infinit dimensionale, vol.I,II, Ed.Academiei R.S.R., 1976, 1979.
3. Doubrovine B., Novikov S., Fomenko A., Geometrie contemporaine, vol.I,II,III, Ed. Mir, 1985.
4. P. Sandovici, M. Tarina, Curs de Geometrie diferentiala, vol.I,II, (lito), Cluj-Napoca, 1974, 1976.
Assessment
Exam.