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Обычная версия сайта
2018/2019

Дифференциальная геометрия

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 5

Course Syllabus

Abstract

The course will serve as an introductory guide to basic topics of Differential geometry: very first introduction to Symplectic and Contact Geometry, the theory of Riemannian manifolds, the theory of affine connections on manifolds, geodesics.
Learning Objectives

Learning Objectives

  • Students will be introduced to the very basic notions of classical differential geometry and then will be given the precise examples of applying characteristic classes, connections and curvature tensors on Riemmanian manifolds.
Expected Learning Outcomes

Expected Learning Outcomes

  • Students would get familiar with with basic notions and instruments of differential geometry, would enhance their methods of solving mathematical problems in various fields. Students would be capable of solving basic problems on differential geometry structures and objects on manifolds.
Course Contents

Course Contents

  • Symplectic and Contact structures. Darboux theorems. Reductions.
  • Differential connection.
  • Parallel transport. Curvature.
  • Affine connection.
  • Introduction to characteristic classes.
  • Riemannian manifold. Levi–Civita connection.
  • Riemannian curvature tensor.
  • Geodesics. The Hopf–Rinow theorem.
  • First and second variation of arc length.
  • Jacobi’s equation and conjugate points.
Assessment Elements

Assessment Elements

  • non-blocking Problem sheets grade
  • non-blocking Final exam
    Written test
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.5 * Final exam + 0.5 * Problem sheets grade
Bibliography

Bibliography

Recommended Core Bibliography

  • John Milnor. (2016). Morse Theory. (AM-51), Volume 51. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1179997

Recommended Additional Bibliography

  • Arnold Neumaier, & Dennis Westra. (2011). Classical and quantum mechanics via Lie algebras. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.D337675E
  • Neumaier, A., & Westra, D. (2008). Classical and Quantum Mechanics via Lie algebras. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsarx&AN=edsarx.0810.1019