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Обычная версия сайта
2019/2020

Научно-исследовательский семинар "Дифференциальная геометрия"

Статус: Дисциплина общефакультетского пула
Когда читается: 3, 4 модуль
Язык: английский
Кредиты: 6

Course Syllabus

Abstract

In this course we present the basic concepts of differential geometry: metric, curvature, connection, etc. The goal of our study is to develop tools for practical efficient computations (including the art of manipulation with indices) supported by a deeper understanding of the geometric meaning of all notions and theorems.
Learning Objectives

Learning Objectives

  • The goal of the course is two-fold: ability of practical computations of differential geometry invariants such that Christoffel symbols, curvature tensor etc., as well as deep geometric understanding of underlying concepts and structures
Expected Learning Outcomes

Expected Learning Outcomes

  • Ability of manipulating with vector and covector fields in arbitrary local coordinates; computation of commutator of fields and exteriour differential of forms.
  • Computation of areas and lengths, finding of principal and Gaussian curvatures of surfaces in the Euclidean three-space
  • Ability to compute the curvature of a given metric on the plane.
  • Learning methods for finding geodesics and their usage in the local and global study of Rienmann manifolds
  • Lerning important examples of Riemann manifolds: sphere, Lobachevski plane in Klein and Poincare models
  • Learning different aspects of the notion of connection and interaction between them. Lerning formalism of manipulating with indices and passing from martix to multiindex notations
  • Understanding of relationship between the local and global invarians of surfaces
Course Contents

Course Contents

  • Vector fields and differential forms
    Manifolds, charts, tangent and cotandent spaces. Vector fields and diffrential forms, coordinate presentation and its transformation under changes of coordinates
  • Differential geometry of surfaces
    The first and the second quadratic forms, principal curvatures, Gaussian curvature
  • Curvature of a plane metric
    Curvature of a plane metric. Theorema egregium.
  • Gauss-Bonnet formula.
    Connection and curvature forms. Parallel translate of vectors on a surface. Local Gauss-Bonnet formula. Global formula.
  • Connection
    Connection as a parralel translate nad connection as covariant derivative. Connection matrix. Guage group transformation. Curvature tensor. Cartan structure equation
  • Riemannian manifolds
    Levi-Cevita connection. Riemann curvature tensor. Ricci tensor and the scalar curvature. Symmetries of the Riemann tensor
  • Geodesics
    Variational interpretation. Exponential map. Normal coordinates. Conjugate points. Global geometry of Riemannian manifolds
Assessment Elements

Assessment Elements

  • non-blocking Midterm test
  • non-blocking Final examination
  • non-blocking Hometasks
Interim Assessment

Interim Assessment

  • Interim assessment (4 module)
    0.6 * Final examination + 0.4 * Midterm test
Bibliography

Bibliography

Recommended Core Bibliography

  • Jean-Pierre Demailly. (2007). Complex analytic and differential geometry. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.495EA558

Recommended Additional Bibliography

  • Differential geometry : connections, curvature, and characteristic classes, Tu, L. W., 2017
  • Spivak, M. (1998). Calculus On Manifolds : A Modern Approach To Classical Theorems Of Advanced Calculus. New York: CRC Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=421137