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Lecturer(s)
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Peška Patrik, RNDr. Ph.D.
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Course content
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1. n-dimensional differentiable manifolds 2. Geometric objects on manifolds (vector fields, tensors) 3. Push-forward and Pull-back, Lie derivative 4. Covariant derivative, manifolds with affine connection 5. Parallel transport and geodesic curves 6. Riemann curvature tensor and Ricci tensor 7. Properties of curvature tensors 8. Riemannian metric, length of curves 9. Geodesics on a Riemannian manifold 10. Spaces of constant curvature, Einstein spaces 11. Isometric and conformal mappings
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Learning activities and teaching methods
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Monologic Lecture(Interpretation, Training), Demonstration, Group work
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Learning outcomes
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The aim of the course is to introduce the fundamentals of Riemannian Geometry and the theory of differentiable manifolds, with emphasis on geometric interpretation of tensor quantities and the relationship between local and global structures. Students will: - understand the concept of differentiable manifolds and geometric objects on them, - master tensor calculus and affine connections, - understand covariant differentiation, parallel transport, and geodesics, - become familiar with curvature tensors (Riemann and Ricci), - understand the role of Riemannian metrics and variational principles, - study spaces of constant curvature and Einstein manifolds, - gain insight into isometric and conformal mappings.
After completing the course, the student: Knowledge: - understands the structure of differentiable manifolds and geometric objects on them, - knows tensor theory and affine connections, - understands the role and properties of the Riemann and Ricci tensors, - understands curvature and spaces of constant curvature. Skills: - is able to perform tensor computations and use covariant derivatives, - works with geodesics and parallel transport, - analyzes curvature using the Riemann tensor, - applies variational principles in geometric problems, - works with isometric and conformal mappings. Competences: - is able to independently solve problems in Differential Geometry, - connects algebraic and geometric methods, - understands deeper links between geometry and physical theories (e.g. general relativity).
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Prerequisites
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Knowledge of Linear Algebra, Mathematical Analysis (especially functions of several variables), and Analytic Geometry is assumed. Basic familiarity with tensor calculus is an advantage.
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Assessment methods and criteria
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Oral exam, Written exam, Student performance
To successfully complete the course, the student must: - demonstrate understanding of the fundamental concepts of Differential Geometry (manifolds, tensors, affine connections, curvature), - actively work with the mathematical formalism (tensor calculus, covariant differentiation), - be able to solve both computational and theoretical problems (e.g. Christoffel symbols, geodesics, curvature tensors), - apply the acquired knowledge to specific geometric situations, - pass the mid-term assessment (if required), - pass the final examination (written and/or oral).
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Recommended literature
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A. Pressley. (2012). Elementary Differential Geometry. Springer.
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DO Carmo M. (1976). Differential Geometry of Curves and Surfaces. Hall.
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. Brno.
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Isham C. J. (1989). Modern Differential Geometry for physicists. World Scientific.
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J. Mikeš, M. Sochor. (2015). Diferenciální geometrie ploch v úlohách. Olomouc.
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Kulhánek Petr. (2016). Obecná relativita. Praha.
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M. A. Akivis, V. V. Goldberg. (1972). An Introduction to Linear Algebra and Tensors. New York.
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M. Umehara, K. Yamada. (2015). Differential Geometry of Curves and Surfaces. World Scientific.
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Mikeš J. et al. (2019). Differential Geometry of Special Mappings. Olomouc.
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Podolský J. (2006). Teoretická mechanika v jazyce diferenciální geometrie. Praha.
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Tahalová, L. (2001). Visual Basic v příkladech. Praha.
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Tapp, K. (2016). Differential Geometry of Curves and Surfaces.
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Tapp Kristopher. (2016). Differential Geometry of curves and surfaces. Switzerland.
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