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
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.
Literature
    required 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
    recommended literature
  • L. Tu. Differential Geometry: Connections, Curvature, and Characteristic Classes. 2017. info
    not specified
  • M. Spivak. Physics for Mathematicians: Mechanics I. 2010. 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
To get course credits it is neccessary to solve three projects assigned by the instructor.

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