#
MU:MU25022 Supp. Tutorial in Dif. Geom II - Course Information

## MU25022 Supplementary Tutorial in Differential Geometry II

**Mathematical Institute in Opava**

Summer 2021

The course is not taught in Summer 2021

**Extent and Intensity**- 0/2/0. 4 credit(s). Type of Completion: zk (examination).
**Teacher(s)**- RNDr. Jiřina Jahnová, Ph.D. (seminar tutor)
**Guaranteed by**- RNDr. Jiřina Jahnová, Ph.D.

Mathematical Institute in Opava **Prerequisites**(in Czech)- TYP_STUDIA ( N )
**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)

**Course objectives**- The goal of the course is a detailed analysis of hands-on approaches to the subject matter presented in the lectures on Differential Geometry II and clarifying less trivial aspects thereof for improving knowledge and skills of students with emphasis on their individual work. Solving certain problems may involve the use of computer algebra software (Maple).
**Syllabus**- Differential forms and Stokes' theorem: motivation for Stokes' theorem, its applications and examples, Lie derivative of differential forms, examples.

Riemannian manifolds, principle, mean and Gaussian curvature and its computation, affine connection, torsion, curvature and Riemannian curvature, connection on a vector bundle, Gauss' Theorema Egregium, examples and computations, covariant derivatives and geodesics, examples.

Hamiltonian classical mechanics: first and second fundamental form on cotangent bundle, hamiltonian vector fields and symplectomorphisms, Poisson bracket, symplectic manifolds, examples and computations, phase space and Hamilton's equations, examples, geometrical optics and Fermat's principle.

- Differential forms and Stokes' theorem: motivation for Stokes' theorem, its applications and examples, Lie derivative of differential forms, examples.
**Literature**- G. F. Torres del Castillo.
*Differentiable Manifolds: A Theoretical Physics Apporach*. 2012. info - L. Krump, J. A. Těšínský, V. Souček.
*Matematická analýza na varietách*. Praha, 1998. info - J. M. Lee.
*Riemannian Manifolds: An introduction to Curvature*. 1997. info

*required literature*- L. Tu.
*Differential Geometry: Connections, Curvature, and Characteristic Classes*. 2017. info

*recommended literature*- M. Spivak.
*Physics for Mathematicians: Mechanics I*. 2010. info

*not specified*- G. F. Torres del Castillo.
**Language of instruction**- Czech
**Further comments (probably available only in Czech)**- The course can also be completed outside the examination period.
**Teacher's information**- To get course credits it is neccessary to solve three projects assigned by the instructor.

- Permalink: https://is.slu.cz/course/sumu/summer2021/MU25022