SUBJECT

Riemannian geometry

lecture + practical

master

Faculty of Science

Mathematics MSc

3

Semesters 1-4

Autumn/Spring semester

The exponential mapping of a Riemannian manifold. Variational formulae for the arc length. Conjugate points. The index form assigned to a geodesic curve. Completeness of a Riemannian manifold, the Hopf-Rinow theorem. Rauch comparison theorems. Non-positively curved Riemannian manifolds, the Cartan-Hadamard theorem. Local isometries between Riemannian manifolds, the Cartan-Ambrose-Hicks theorem. Locally symmetric Riemannian spaces.

Submanifold theory: Connection induced on a submanifold. Second fundamental form, the Weingarten equation. Totally geodesic submanifolds. Variation of the volume, minimal submanifolds. Relations between the curvature tensors. Fermi coordinates around a submanifold. Focal points of a submanifold.

• M. P. do Carmo: Riemannian geometry. Birkhäuser, Boston, 1992.
• J. Cheeger, D. Ebin: Comparison theorems in Riemannian geometry. North-Holland,  Amsterdam 1975.

Further reading:

• S. Gallot, D. Hulin, J. Lafontaine: Riemannian geometry. Springer-Verlag, Berlin, 1987.