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Course module: NWI-WB045B
NWI-WB045B
Riemannian Geometry
Course infoSchedule
Course moduleNWI-WB045B
Credits (ECTS)6
CategoryBA (Bachelor)
Language of instructionDutch
Offered byRadboud University; Faculty of Science; Wiskunde, Natuur- en Sterrenkunde;
Lecturer(s)
Coordinator
dr. A.Y. Burtscher
Other course modules lecturer
Lecturer
dr. A.Y. Burtscher
Other course modules lecturer
Contactperson for the course
dr. A.Y. Burtscher
Other course modules lecturer
Examiner
dr. A.Y. Burtscher
Other course modules lecturer
Academic year2022
Period
KW3-KW4  (30/01/2023 to 31/08/2023)
Starting block
KW3
Course mode
full-time
Remarks-
Registration using OSIRISYes
Course open to students from other facultiesYes
Pre-registrationNo
Waiting listNo
Placement procedure-
Aims
The student knows how to use the following notions, both in theoretical proofs, as well as in explicit computations:
  • smooth vector bundle
  • Riemannian manifolds 
  • connections on vector-bundles and the Levi-Civita connection
  • geodesics
  • parallel transport
  • the various notions of curvature on a Riemannian manifold
  • Jacobi fields
Content
Riemannian geometry was born in Bernhard Riemann's famous habilitation lecture on the foundations (hypotheses) of geometry in 1854. By defining a Riemannian metric on a manifold as a smoothly varying inner product on the tangent space at any point, we can introduce local concepts such as length, angle, area, volume and curvature, and thus we can extend concepts from classical Gaussian differential geometry (i.e., surfaces in a three-dimensional Euclidean space) to abstract (i.e., no longer embedded in a certain Euclidean space) manifolds of arbitrary dimension. Global geometric quantities can be obtained using integration.

This introductory course covers all basic notions of Riemannian geometry. It is strongly recommended to students interested in geometry and/or mathematical physics. For instance, the indefinite variation of Lorentzian geometry is the language needed to understand Einstein's general theory of relativity about the structure of our universe.
Level

Presumed foreknowledge
The course Manifolds in the previous semester is an absolute must, without which this course will be incomprehensible. Useful but not absolute necessary is the second year course on Curves and Surfaces in a three dimensional Euclidean space.
Test information
Written or oral exam
Specifics

Recommended materials
Book
Title:Riemannian geometry
Author:Manfredo Perdigão do Carmo
Publisher:Birkhäuser
Book
ISBN:978-3-319-91754-2
Title:Introduction to Riemannian manifolds
Author:John M. Lee
Publisher: Graduate Texts in Mathematics 176, Springer
Book
ISBN:978-3-319-26652-7
Title:Riemannian geometry
Author:Peter Petersen
Publisher:Graduate Texts in Mathematics 171, Springer
Instructional modes
Course
Attendance MandatoryYes

Tests
Final result
Test weight1
Test typeExam
OpportunitiesBlock KW4, Block KW4

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Kies de Nederlandse taal