Lecturer(s)


Mikeš Josef, prof. RNDr. DrSc.

Course content

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 PetersonCodazziMainardi 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 on differential geometry on curves, surfaces and manifold.
1. Knowledge Describe properties of the differential geometry of curves and surfaces and manifolds.

Prerequisites

Knowledge of principles of analytical geometry.

Assessment methods and criteria

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.

Recommended literature


Budinský B. Kepr B. (1970). Základy diferenciální geometrie s technickými aplikacemi. SNTL Praha.

Doupovec, M. (1999). Diferenciální geometrie a tenzorový počet. VUT Brno.

Gray A. (1994). Differential geometry. CRC Press Inc.

Metelka, J. (1969). Diferenciální geometrie. SPN Praha.

Oprea, J. (2007). Differential geometry and its aplications. MAA Pearson Educ.

Pogorelov, A. V. (1969). Diferencialnaja geometrija.. Nauka Moskva.

Vanžurová, A. (1996). Diferenciální geometrie křivek a ploch. UP Olomouc.
