MU25009 Chapters in Differential Geometry

Mathematical Institute in Opava
Summer 2019
Extent and Intensity
2/2/0. 6 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Michal Marvan, CSc. (lecturer)
RNDr. Hynek Baran, Ph.D. (seminar tutor)
Guaranteed by
doc. RNDr. Michal Marvan, CSc.
Mathematical Institute in Opava
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives (in Czech)
V tomto předmětu budou probrány další partie klasické i moderní diferenciální geometrie, ve kterých by měl absolvent navazujícího magisterského studia diferenciální geometrie dosáhnout základní orientace. Obsah může reflektovat zájmy posluchačů.
Syllabus
  • 1. Hypersurfaces in Euclidean spaces: second fundamental form, Gauss-Weingarten equations, Gauss-Mainardi-Codazzi equations, Bonnet's theorem, normal sections, principal curvatures, principal coordinates, mean and Gaussian curvature, theorema egregium, normal congruences, focal hypersurfaces, Gauss map, third fundamental form.
    2. Minimal surfaces, pseudospheric surfaces, models of Lobachevsky geometry.
    3. Complex manifolds, complex structures on a real manifold, complex differential forms, holomorphic forms, Kaehler manifolds, Calabi-Yau manifolds, applications in string theory.
    4. Basic theory of elliptic curves and elliptic functions.
    5. Contact structures, a nonlinear partial differential equation of first order and its solution.
Literature
    recommended literature
  • S. P. Novikov, I. A. Taimanov. Modern Geometric Structures and Fields. Amer. Math. Soc., 2006. info
  • A. M. Vinogradov, I. S. Krasilshchik. Symmetries And Conservation Laws for Differential Equations in Mathematical Physics. Amer. Math. Soc., 1999. info
  • V. V. Prasolov, Yu. P. Solovev. Elliptic Functions and Elliptic Integrals. Amer. Math. Soc., 1997. info
Language of instruction
Czech
Further Comments
The course can also be completed outside the examination period.
The course is also listed under the following terms Summer 2013, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2018.
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