# MU:MU03038 Differential Geometry I - Course Information

## MU03038 Differential Geometry I

**Mathematical Institute in Opava**

Winter 2016

**Extent and Intensity**- 2/2/0. 6 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**- prof. RNDr. Artur Sergyeyev, Ph.D., DSc.

Mathematical Institute in Opava **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, N1101)
- Mathematical Analysis (programme MU, M1101)
- 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, 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**- - 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**- 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 - C. Isham.
*Modern Differential Geometry for Physicists*. Singapore, 1999. 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*- S. Caroll.
**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.

- Enrolment Statistics (Winter 2016, recent)
- Permalink: https://is.slu.cz/course/sumu/winter2016/MU03038