MU03039 Differential Geometry II

Mathematical Institute in Opava
Summer 2019
Extent and Intensity
4/2/0. 8 credit(s). Type of Completion: zk (examination).
Teacher(s)
prof. RNDr. Artur Sergyeyev, Ph.D., DSc. (lecturer)
RNDr. Petr Vojčák, Ph.D. (seminar tutor)
Guaranteed by
RNDr. Petr Vojčák, Ph.D.
Mathematical Institute in Opava
Prerequisites
MU03038 Differential Geometry I
MU/03038
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
Differential geometry is the part of geometry, which makes use of the methods from calculus for the study of curves, (hyper) surfaces, etc. In its study of geometrical objects, differential geometry concentrates on the so-called invariant properties which do not depend on the choice of coordinate systems. Differential geometry is mainly concerned with local properties of geometrical objects, that is, the properties of sufficiently small parts of those objects.
Syllabus
  • 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)
Literature
    recommended literature
  • S. Caroll. Lecture Notes on General Relativity. URL info
  • D. Krupka. Matematické základy OTR. info
  • M. Fecko. Diferenciálna geometria a Lieove grupy pre fyzikov. Bratislava, Iris, 2004. info
  • M. Wisser. Math 464: Notes on Differential Geometry. 2004. URL info
  • C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999. info
  • O. Kowalski. Úvod do Riemannovy geometrie. Univerzita Karlova, Praha, 1995. info
  • B. A. Dubrovin, A. T. Fomenko, S. P. Novikov. Modern Geometry - Methods and Applications, Parts I and II,. Springer-Verlag, 1984. info
  • F. Warner. Foundations of differentiable manifolds and Lie groups. Springer-Verlag, N.Y.-Berlin, 1971. info
  • M. Spivak. Calculus on Manifolds. 1965. info
    not specified
  • John M. Lee. Introduction to Smooth Manifolds. 2006. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
Oral exam; further requirements to be specified in the course of the semester.
The course is also listed under the following terms Winter 1997, Summer 1998, Winter 1998, Summer 1999, Summer 2000, Summer 2001, Summer 2002, Summer 2003, Summer 2004, Summer 2005, Summer 2006, Summer 2007, Summer 2008, Summer 2009, Summer 2010, Summer 2011, Summer 2012, Summer 2013, Summer 2014, Summer 2015, Summer 2016, Summer 2017, Summer 2018, Summer 2021.
  • Enrolment Statistics (Summer 2019, recent)
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