Dear students, Due to Coronavirus Pandemie we will switch on on-line teaching.

During Lecture (Monday 20 April 10 am) we will recall the parallel transport and geodesics. Our lecture will be founded on lecture notes and these slides

During Lecture and Tutorial (Thursday 23 10 am--11.50 am) we will formulate Theorem about changing of a vector during parallel transport along closed curve on the surface in $E^3$. Then we will exlain what is it Weingarten (Shape) operator, and how to calculate Gaussian curvature in terms of Shape operator. We will end the lecture calculating Gaussian curvature for cylinder and sphere Our lecture will be founded on lecture notes and these slides

During these two hours we will find a time and I will explain how to calculate Christoffel symbols of Levi-Civita connection using Euler-lagrange equations of motion. Our improvised tutorial will be founded on homeworks and these slides

During Lecture tomorrow (on Monday 27April 10 am) we will consider again the Theorem about parallel transport of vector over closed curve, and we consider more detailed formulation of this theorem. Then we deduce from this statement Gauss Theorema Egregium. lecture will be founded on lecture notes and these slides

During Lecture and tutorial tomorrow (on Thursday 30 April 10-11) we will define curvature tensor for arbitrary connection, express it in terms of Christoffel symbols. The fact that curvature is expressed via tensor implies that we can answer the question: does there exist coordinates such that metric is Euclidean. the lecture will be founded on lecture notes and these slides

During Lecture tomorrow (on Monday 04 May 10-11) we will consider Gauss-Bonet Theorem about integral of Gaussain curvature over surface and its relation to Euler characteristic of the surface. the lecture will be founded on lecture notes and these slides

During Lecture and tutorial tomorrow (on Thursday 08 May 10-12) we will consider Gauss-Bonet Theorem and Euler characteristic. We will try to consider Euler-chracteristic in terms of homology (cohomology) theory.The lecture will be founded on lecture notes and these slides. It will be the LAST lecture.