General Theory of Relativity or the theory of relativistic gravitation is the one which describes black holes, gravitational waves and expanding Universe. The goal of the course is to introduce you into this theory. The introduction is based on the consideration of many practical generic examples in various scopes of the General Relativity. After the completion of the course you will be able to solve basic standard problems of this theory. We assume that you are familiar with the Special Theory of Relativity and Classical Electrodynamics. However, as an aid we have recorded several complementary materials which are supposed to help you understand some of the aspects of the Special Theory of Relativity and Classical Electrodynamics and some of the calculational tools that are used in our course. Also as a complementary material we provide the written form of the lectures at the website: https://math.hse.ru/generalrelativity2015

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Covariant differential and Riemann tensor

We start with the definition of what is tensor in a general curved space-time. Then we define what is connection, parallel transport and covariant differential. We show that for Riemannian manifolds connection coincides with the Christoffel symbols and geodesic equations acquire a clear geometric meaning. We end up with the definition of the Riemann tensor and the description of its properties. We explain how Riemann tensor allows to distinguish flat space-time in curved coordinates from curved space-times. For this module we provide complementary video to help students to recall properties of tensors in flat space-time.