Course: Geometry II

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Course title Geometry II
Course code KAG/KGEII
Organizational form of instruction Seminar
Level of course Master
Year of study not specified
Semester Winter
Number of ECTS credits 7
Language of instruction Czech
Status of course unspecified
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Mikeš Josef, prof. RNDr. DrSc.
Course content
1. Afinne mappings: Definition and properties. Associated homomorfisms. The determination theorem. Analytic form. 2. Group of afinne transformations: Modul of affinity, equiafinity. Self-conjugate points and directions. Homothetic affinity translation and homothety. 3. Fundamental affinity and their meaning. Classification of affinity in planes. 4. Isometric mappings: Definition and properties. Analytic form. Group of izometry. Symmetry with respect to a hyperplain. 5. Classification of izometry on 1-, 2-, 3-dimensional Euclidian spaces. 6. Similarity mappings: Definition and properties. Analytic form. Group of similarity. Decomposition of similarity into isometry and homothety. Using similarity for solution of constructive problems and proofs. Construction of a center of similarity in a plane. 7. Potency of a point in a circle. Chordal of two circles. Bundle of circles. Apollonius and Papp's problems. 8. Cyclic mappings: Cyclic inversion in the Möbius plane. Mappings of cyclic curves. Using cyclic inversions for solution of constructive problems. 9. Transformation of Euclidean plain in complex coordinates. Analytic form of affine, isometric and similarity mappings. 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.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming)
Learning outcomes
Describe principles and classification of affine mappings. Describe principles on differential geometry on curves, surfaces and manifold.
1. Knowledge Describe properties of affine mappings on affine spaces. Describe properties of the differential geometry of curves and surfaces and manifolds.
Knowledge of affine and Euklidean spaces.

Assessment methods and criteria
Oral exam, Written exam

Credit: the student has to participate actively in seminars and pass a written test.
Recommended literature
  • Berger, M. (1987). Geometry I, II. Universitext Springer-Verlag Berlin.
  • Boček L. Sekanina M. (1988). Geometrie II. SPN Praha.
  • Budinský B., Kepr B. (1970). Základy diferenciální geometrie s technickými aplikacemi. SNTL Praha.
  • Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.
  • Jachanová, Marková, Žáková. (1989). Geometrie II. VUP Olomouc.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester