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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. Vector functions. 2. Parametrization of curves. Orientation. Methods of determination of curves. 3. Length of a curve, natural parameters. 4. Tangents, planes of osculatory, the Frenet frame. 5. The Frenet formulas, curvature, torsion. Natural equation of a curve. 6. Joint of curves, circle of osculatory. 7. Parametrization of surfaces. Methods of determination of surfaces. 8. Tangents. Tangent planes and normals of a surface. Orientation of surfaces. 9. First and second fundamental form of a surface anf their purpose. 10. The Meussnier formulas and theorem. 11. Principal directions. Normal, geodetic, principal, medium and Gauss curvatures. Euler's formula. 12. Gauss and Weiengarten formulas. 13. Gauss and Peterson-Codazzi-Mainardi formulas. Christoffel symbols. 14. The Egregium theorem. 15. Special curves on surfaces. 16. Special surfaces (set surfaces, surface of a constant curvature, surfaces of revolution). 17. Differentiable manifolds, affine connections, the Riemann manifolds.
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
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Learning outcomes
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Describe principles on differential geometry on curves, surfaces and manifold.
1. Knowledge Describe properties of the differential geometry of curves and surfaces and manifolds.
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Prerequisites
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Knowledge of principles of analytical geometry.
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Assessment methods and criteria
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Oral exam, Written exam
Credit: the student has to participate actively in seminars and pass the test. Exam: oral. The student has to understand the subject and be able to prove the principal results.
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Recommended literature
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Gray A. (1994). Differential geometry. CRC Press Inc.
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J. Mikeš, E. Stepanova, A. Vanžurová et al. (2015). Differential geometry of special mappings. UP Olomouc.
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J. Mikeš, M. Sochor. (2013). Diferenciální geometrie ploch v úlohách. UP OLomouc.
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Kolář, I., Pospíšilová, L. (2007). Diferenciální geometrie křivek a ploch. El. publ. MU Brno.
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Metelka, J. (1969). Diferenciální geometrie. SPN Praha.
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Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.
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Pogorelov, A. V. (1969). Diferencialnaja geometrija.. Nauka Moskva.
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