Course: Selected Lessons in Riemannian Geometry

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Course title Selected Lessons in Riemannian Geometry
Course code KAG/GVRG9
Organizational form of instruction Lecture
Level of course Master
Year of study not specified
Semester Winter
Number of ECTS credits 2
Language of instruction Czech
Status of course Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Mikeš Josef, prof. RNDr. DrSc.
Course content
1. n-dimensional differentiable manifolds. 2. Tensors on manifolds. 3. Manifolds with affine connection, covariant derivation. 4. Parallel transport. Geodetic curves. 5. Riemannian and Ricci tensors. 6. Riemannian metrics, length of curves. 7. Geodetic curves on Riemannian space. 8. Properties of Riemannian and Ricci tensors. 9. Sectional curvature on Riemannian space. 10. Spaces of constant curvature, Einstein spaces. 11. Isometric and conformal mappings.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
Learning outcomes
Understand theory of Riemannian spaces.
2. Comprehension Explain the conception of the Riemannian geometry
Prerequisites
Principles of the analytical geometry.

Assessment methods and criteria
Oral exam, Written exam

Credit: active participation.
Recommended literature
  • Conlon L. (1993). Differentiable manifolds: a first course. Boston, Basel, Berlin, Birkhauser.
  • Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
  • Eisenhart, L.P. (2000). Non-Riemannian Geometry. Amer. Math. Soc. Colloquium Publ. 8.
  • Gromol D. Klingenberg V., Meyer V. (1980). Riemannova geometrija v celom. Nauka Moskva.
  • Kowalski, O. (1995). Úvod do Riemannovy geometrie. Praha.
  • Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
  • Pogorelov, A. V. (1969). Diferencialnaja geometrija.. Nauka Moskva.
  • Poznyak, E. G., Shikin, E. V. (1990). Differential geometry. The first acquaintance (Russian). Izdatel'stvo Moskovskogo Universiteta Moskva.
  • Sinyukov, N. S. (1979). Geodesic mappings of Riemannian spaces. Nauka Moskva.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Teaching Training in Mathematics for Secondary Schools (2015) Pedagogy, teacher training and social care 2 Winter
Faculty of Science Discrete Mathematics (2015) Mathematics courses 2 Winter
Faculty of Science Teaching Training in Descriptive Geometry for Secondary Schools (2015) Pedagogy, teacher training and social care 2 Winter