MSC in Mathematics, Pure Mathematics, Elective Course


 Riemannian Geometry II


 Dr. Balázs Csikós (associate professor)

 Dr. Gyula Lakos (assistant)

 Dr. Gábor Moussong (assistant professor)

 Dr. László Verhóczki (associate professor)




 2nd semester


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. Cartan-Hadamard theorem. Focal points. Morse index form and causality. Causality relations on time-oriented Lorentz manifolds. Achronal and acausal sets. Cauchy hypersurfaces. Cauchy development. Space-like hypersurfaces. Cauchy horizon. Hawking’s singularity theorem.     


J. K. Beem – P. E. Ehrlich – K. L. Easley: Global Lorentzian geometry. 2nd ed. Marcel Dekker, New York, 1996.

S. W. Hawking – G. F. R. Ellis: The large scale structure of space-time. Cambridge University Press, Cambridge, 1973.

B. O’Neill: Semi-Riemannian geometry. Academic Press, New York, 1983.

form of tuition

Two hours of lecture per week.

mode of assessment

 oral exam