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The goal of this lecture is to acquaint the students with one of the basic techniques of the Mathematical Physics, namely pseudo-Riemannian geometry.
Last update: Kowalski Oldřich, prof. RNDr., DrSc. (10.09.2013)
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The goal of the lecture is to acquaint students with one of the basic structures of differential geometry, namely with a smooth manifold equipped with a Riemannian metric tensor and its connection. Last update: Golovko Roman, Ph.D. (26.09.2018)
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There will be several homeworks. As a requirement to take the final exam students must submit solutions to at least one homework. The final exam will be an oral exam. Last update: Golovko Roman, Ph.D. (13.09.2021)
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1) O. Kowalski, Základy Riemannovy geometrie, skripta, 2. vydání, vydavatelství Karolinum, 2001.
2) P. do Carmo, Riemannian Geometry 1, Birkhaeuser.
3) M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1 - 2., Publish or Perish Inc.
4) P. Petersen, Riemannian Geometry, Springer, Graduate Texts in Mathematics, Vol. 171, 2nd Edition, 2006.
5) W. Curtis, F. Miller, Differential Manifolds & Theoretical Physics, Pure and Applied Math.
6) S. Kobayashi and K. Nomizu, Foundations of Differential geometry I, II, Interscience Publishers 1963, 1969.
7) S. Helgason, Differencial´naja geometrija i simmetričeskije prostranstva (překlad z angličtiny), Izd. MIR, Moskva 1964 (Kapitola 1).
8) S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic press, 1978.
9) R. L. Bishop, R. J. Crittenden, Geometry of Manifolds, AMS Chelsea Publishing, 2001. Last update: Golovko Roman, Ph.D. (26.09.2018)
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The method of teaching is the standard lecture and exercise sessions. The individual study is also possible. Last update: Golovko Roman, Ph.D. (26.09.2018)
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For the oral part of the exam it is necessary to know the whole content of lecture. You will get time to write a preparation for the oral part which the knowledge of definitions, theorems and their proofs is tested. We test as well the understanding to the lecture, you will have to prove an easy theorem which follows from statements from the lecture.
Last update: Golovko Roman, Ph.D. (26.09.2018)
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Basic notions from general topology. Topological and differentiable manifolds, maps between manifolds. Submanifolds in the Euclidean space. Tangent spaces, tangent maps, vector fields, Lie bracket of vector fields. Affine connection on a manifold as differentiation of vector fields. The Levi-Civita connection on a manifold in R^n. The parallel transport along curves, geodesic curves - definitions and existence theorems. Exponential map at a point. The torsion tensor field and the curvature tensor field, its geometric meaning. Riemannian (pseudo-Riemannian) metric, the induced structure of a metric space. The Riemannian connection - existence and uniqueness, relationship with the Levi-Civita connection (on a submanifold with induced metric). The Gaussian formula and its geometric interpretation for surfaces - Gauss theorem. The Gauss curvature of a surface. The sectional curvature of a Riemannian manifold, spaces with constant curvature. Extremal properties of geodesics. Global properties of geodesics on a complete Riemannian manifold.
Last update: Golovko Roman, Ph.D. (26.09.2018)
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Basics of topology (finer, coarser, induced topology) and of calculus of functions of several variables. Last update: Golovko Roman, Ph.D. (26.09.2018)
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