Course Content (Syllabus)
The notion of the metric and of the isometry map. Theory of connections. the notion of the covariant derivative. Levi-Civita connection. Geodesics. Curvature tensor. Sectional curvature, Ricci curvature and scalar curvature. Hypersurfaces of a Riemannian manifold. Riemannian manifolds of constant curvature.
Elements from the theory of smooth manifolds. Riemannian manifolds. Linear connections. Geodesics and curvature. Sectional curvature. Jacobi fields and second fundamental form.
Course Bibliography (Eudoxus)
Δημητρίου Κουτρουφιώτη, Διαφορική Γεωμετρία, Πανεπιστήμιο Ιωαννίνων 1994.
Ανδρέα Αρβανιτογεώργου, Γεωμετρία Πολλαπλοτήτων – Πολλαπλότητες Riemann και Ομάδες Lie, Ελληνικά Ακαδημαϊκά Ηλεκτρονικά Συγγράμματα και Βοηθήματα, www.kallipos.gr
Additional bibliography for study
M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992.
John M. Lee, Riemannian manifolds. An introduction to curvature, GTM 176, Springer-Verlag 1997.
Andrew Mclnerney, First Steps in Differential Geometry, Riemannian, Contact, Symplectic, Springer 2013.
Leonor Godinho, Jose Natario, An Introduction to Riemannian Geometry, With Applications to Mechanics and Relativity, Springer 2014.
Loring W. Tu, An introduction to Manifolds, Universitext, Springer 2011.