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MU:MU03039 Differential Geometry II - Course Information

## MU03039 Differential Geometry II

**Mathematical Institute in Opava**

Summer 2021

**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 **Timetable**- Tue 16:25–18:00 R1, Thu 18:05–19:40 R1
- Timetable of Seminar Groups:

*J. Vašíček* **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**- Geometry and Global Analysis (programme MU, NMgr-M)
- Geometry and Global Analysis (programme MU, N1101)
- Mathematical Analysis (programme MU, N1101)
- Mathematics (programme MU, B1101)

**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)

- Differential forms -- continued (orientability, integration on manifolds, the Stokes theorem and its consequences)
**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

*required 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

*recommended literature*- P. Krtouš.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- Study Materials

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.

- Enrolment Statistics (recent)

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