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MU:MU03038 Differential Geometry I - Course Information

## MU03038 Differential Geometry I

**Mathematical Institute in Opava**

Winter 2020

**Extent and Intensity**- 2/2/0. 6 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 16:25–18:00 112
- Timetable of Seminar Groups:

*J. Vašíček* **Prerequisites**(in Czech)- TYP_STUDIA ( BN )
**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**- Applied Mathematics (programme MU, B1101)
- 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.
**Syllabus**- - Smooth manifolds (definition, coordinate systems, atlases, submanifolds, examples of manifolds, mappings of manifolds)

- Tangent and cotangent space to the manifold and their relationship

(definitions and properties, tangent vectors of curves, tangential views, and kotečný tangent bundles)

- Vector fields on manifolds and their properties

(different definitions of a vector field and their relations, the Lie bracket and its properties, F-related vector fields and their properties, one-parameter groups, flows and integral curves and their relations)

- Differential forms on manifolds and their properties

(definition of differential forms; pullback, the exterior product, Lie derivative, exterior derivative, contraction and their relations and properties)

- - Smooth manifolds (definition, coordinate systems, atlases, submanifolds, examples of manifolds, mappings of manifolds)
**Literature**- D. Krupka.
*Matematické základy OTR*. info - C. Isham.
*Modern Differential Geometry for Physicists*. Singapore, 1999. info - O. Kowalski.
*Úvod do Riemannovy geometrie*. Univerzita Karlova, Praha, 1995. info

*required literature*- F. Larusson.
*Differential Geometry*. Adelaide. URL info - S. Caroll.
*Lecture Notes on General Relativity*. URL info - John M. Lee.
*Introduction to Smooth Manifolds*. 2006. info - M. Fecko.
*Diferenciálna geometria a Lieove grupy pre fyzikov*. Bratislava, Iris, 2004. info - J. Musilová, D. Krupka.
*Integrální počet na Euklidových prostorech a diferencovatelných varietách*. SPN, Praha, 1982. info - M. Spivak.
*Calculus on Manifolds*. 1965. info

*recommended literature*- D. Krupka.
**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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