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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)
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Learning outcomes
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The aim of the course is to acquaint students with the differential geometry on the manifolds. The student will learn the basics of tensor calculus and analysis, which appears in this topic as a necessary apparatus.
Explains the concept of Reimann's geometry.
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Prerequisites
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Basic knowledge of integral and differential calculus and analytical geometry is assumed.
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Assessment methods and criteria
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Student performance
Credit: write two credit-based written papers, each with at least half the points. Exam: basic understanding of the material, ability to apply in examples.
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Recommended literature
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A. Pressley. (2012). Elementary Differential Geometry. Springer- Verlag.
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DO Carmo M. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall.
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Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT 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. Dover Publications, 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. UK Praha.
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Tahalová, L. (2001). Visual Basic v příkladech. Praha : BEN, 191 s.
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Tapp, K. (2016). Differential Geometry of Curves and Surfaces. Springer.
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Tapp Kristopher. (2016). Differential Geometry of curves and surfaces. Switzerland.
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