MSC in Mathematics, Pure Mathematics, Elective Course


 Riemannian Geometry I


 Dr. Balázs Csikós (associate professor)

 Dr. Gyula Lakos (teaching assistant)

 Dr. Gábor Moussong (assistant professor)

 Dr. László Verhóczki (associate professor)




 1st semester


The notion of semi-Riemannian, Riemannian, and Lorentz manifolds. Levi-Cività 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. Schouten-Struik theorem. Kulkarni-Nomizu product. The decomposition of the curvature tensor. Singer-Thorpe theorem. Weyl tensor. Time-cones of Lorentz vector spaces. Extremal properties of time-like geodesics of Lorentz manifolds. Einstein equation. The Schwarzschild space-time.


R. L. Bishop – R. J. Crittenden: Geometry of manifolds. American Mathematical Society, New York, 1964.

S. Kobayashi – K. Nomizu: Foundations of differential geometry I—II, Wiley, New York, 1963, 1969.

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