Course: Differential Geometry II

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Course title Differential Geometry II
Course code MU/03039
Organizational form of instruction Lecture + Lesson
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 8
Language of instruction Czech
Status of course Compulsory, Compulsory-optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Course availability The course is available to visiting students
Lecturer(s)
  • SERGYEYEV Artur, doc. RNDr. Ph.D.
  • VOJČÁK Petr, RNDr. Ph.D.
Course content
Differential forms - continued (orientability, integration on manifolds, the Stokes theorem and its consequences) Tensor fields on manifolds and their properties (definition, operations on tensors, including symmetrization, antisymmetrization, tensor product, the Lie derivative) Affine connections and related issues (the torsion tensor, the curvature tensor, parallel transport of vectors, geodesics, covariant derivatives, geometrical meaning of the curvature tensor) Manifolds with the metric ((pseudo) Riemannian manifolds, Levi-Civita connection, curvature tensor, Ricci tensor, scalar curvature, isometries and the Killing equation, integrating functions on manifold with a metric, the Levi-Civita (pseudo)tensor, volume element, Hodge duality). Basics of the Lie groups theory (the definition of the Lie group, left- and right-invariant vector fields and differential forms and their properties, the Lie algebra and its relationship with the Lie group)

Learning activities and teaching methods
unspecified
Recommended literature
  • B. A. Dubrovin, A. T. Fomenko, S. P. Novikov. Modern Geometry - Methods and Applications, Parts I and II,. Springer-Verlag, 1984.
  • C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999.
  • D. Krupka. Matematické základy OTR.
  • F. Warner. Foundations of differentiable manifolds and Lie groups. Springer-Verlag, N.Y.-Berlin, 1971.
  • John M. Lee. Introduction to Smooth Manifolds. 2006.
  • M. Fecko. Diferenciálna geometria a Lieove grupy pre fyzikov. Bratislava, Iris, 2004.
  • M. Spivak . Calculus on Manifolds. 1965.
  • M. Wisser. Math 464: Notes on Differential Geometry. 2004.
  • O. Kowalski. Úvod do Riemannovy geometrie. Univerzita Karlova, Praha, 1995.
  • S. Caroll. Lecture Notes on General Relativity.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Mathematical Institute in Opava Mathematics (2014-IVT) Mathematics courses 3 Summer
Mathematical Institute in Opava Mathematics (2014-F) Mathematics courses 3 Summer
Mathematical Institute in Opava Mathematics (2-F) Mathematics courses 3 Summer
Mathematical Institute in Opava Geometry and Global Analysis (1) Mathematics courses 1 Summer
Mathematical Institute in Opava Mathematical Analysis (1) Mathematics courses 2 Summer
Mathematical Institute in Opava Mathematics (1-IVT) Mathematics courses 3 Summer
Mathematical Institute in Opava Mathematical Analysis (1-IVT) Mathematics courses 5 Summer