discipline 
MSC in Mathematics, Pure Mathematics,
Elective Course

subject 
Riemannian Geometry
I

lecturers 
Dr. Balázs Csikós (associate professor) Dr. Gyula Lakos (teaching assistant) Dr. Gábor Moussong (assistant professor) Dr. László Verhóczki (associate professor) 
credits 
2 
period 
1st semester 
curriculum 
The notion of semiRiemannian, Riemannian, and
Lorentz manifolds. LeviCività covariant derivative. Parallel
transport. Geodesics. Torsion and curvature of covariant derivatives. Bianchi
identities. Geometry of submanifolds. The second fundamental form. Geometry
of warped products. Sectional curvature. Manifolds of constant curvature.
Schur’s theorem. Ricci tensor, scalar curvature. Einstein manifolds.
SchoutenStruik theorem. KulkarniNomizu product. The decomposition of the
curvature tensor. SingerThorpe theorem. Weyl tensor. Timecones of Lorentz
vector spaces. Extremal properties of timelike geodesics of Lorentz
manifolds. Einstein equation. The Schwarzschild spacetime. 
literature 
R. L. Bishop – R. J. Crittenden: Geometry of
manifolds. American Mathematical Society, S. Kobayashi – K. Nomizu: Foundations of
differential geometry I—II, Wiley, B. O’Neill: SemiRiemannian geometry. Academic
Press, 
form of tuition 
Two hours
of lecture per week. 
mode of assessment 
oral exam 