discipline 
MSC in Mathematics, Pure Mathematics,
Elective Course

subject 
Riemannian Geometry
II

lecturers 
Dr. Balázs Csikós (associate professor) Dr. Gyula Lakos (assistant) Dr. Gábor Moussong (assistant professor) Dr. László Verhóczki (associate professor) 
credits 
2 
period 
2nd semester 
curriculum 
Sprays. Exponential map. Normal neighbourhoods.
Completeness of Riemann and Lorentz manifolds. The first and second variation
of arc length, Synge’s formula. Jacobi fields. Morse index form, conjugate
points. Local extremal values of geodesics. CartanHadamard theorem. Focal
points. Morse index form and causality. Causality relations on timeoriented
Lorentz manifolds. Achronal and acausal sets. Cauchy hypersurfaces. Cauchy
development. Spacelike hypersurfaces. Cauchy horizon. Hawking’s singularity
theorem. 
literature 
J. K. Beem – P. E. Ehrlich – K. L. Easley: Global
Lorentzian geometry. 2^{nd} ed. Marcel Dekker, S. W. Hawking – G. F. R. Ellis: The large scale
structure of spacetime. B. O’Neill: SemiRiemannian geometry. Academic
Press, 
form of tuition 
Two hours
of lecture per week. 
mode of assessment 
oral exam 