## MU03039 Differential Geometry II

Mathematical Institute in Opava
Summer 2022
Extent and Intensity
4/2/0. 8 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Artur Sergyeyev, DSc. (lecturer)
Mgr. Jakub Vašíček (seminar tutor)
Guaranteed by
doc. RNDr. Artur Sergyeyev, DSc.
Mathematical Institute in Opava
Prerequisites
MU03038 Differential Geometry I && TYP_STUDIA ( BN )
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, and more generally of the so-called manifolds. 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. The purpose of the course is introducing the students to the basics of differential geometry. In the present course, which is a continuation of Differential geometry I, we shall mostly deal with tensor calculus on manifolds and with Lie groups.
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 and torsion tensors)
Manifolds with the metric ((pseudo) Riemannian manifolds, Levi-Civita connection,
curvature tensor, Ricci tensor, scalar curvature, isometries and the Killing equation,
integration on manifold with a metric, the Levi-Civita (pseudo)tensor, volume element, basics of the 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 relation to the Lie group)
Literature
required literature
• P. Krtouš. Geometrické metody ve fyzice. Praha. URL info
• John M. Lee. Introduction to Smooth Manifolds. 2006. info
• C. Isham. Modern Differential Geometry for Physicists. Singapore, 1999. info
• O. Kowalski. Úvod do Riemannovy geometrie. Univerzita Karlova, Praha, 1995. info
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
• 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
Language of instruction
Czech
Further comments (probably available only in Czech)
The course can also be completed outside the examination period.
Teacher's information
Attending the lectures is desirable. In the course of the first lecture the lecturer will communicate to the students the requirements regarding the conditions of successfully passing the subject. Succeeding in the final test requires attaining the score of at least 60 percent at the test papers (typically two in the semester) or 70 percent at the remedial final test. The exact requirements and dates for submitting the papers are set by the tutor. The exam is oral. There the knowledge and the skills of the students gained during the course in question will be checked. Passing final test is required for admission to the exam.
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 2019, Summer 2021.
• Enrolment Statistics (Summer 2022, recent)