## 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
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)
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
• 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
not specified
• John M. Lee. Introduction to Smooth Manifolds. 2006. info
• M. Wisser. Math 464: Notes on Differential Geometry. 2004. URL info
• O. Kowalski. Úvod do Riemannovy geometrie. Univerzita Karlova, Praha, 1995. 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, Winter 1999, Winter 2000, Winter 2001, Winter 2002, Winter 2003, Winter 2004, Winter 2005, Winter 2006, Winter 2007, Winter 2008, Winter 2009, Winter 2010, Winter 2011, Winter 2012, Winter 2013, Winter 2014, Winter 2015, Winter 2017, Winter 2018, Winter 2019, Winter 2020, Winter 2021, Winter 2022.
• Enrolment Statistics (Winter 2016, recent)