2019/2020

Research Seminar "Differential Geometry"

Category 'Best Course for Career Development'

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Maxim Kazaryan

Language: English

ECTS credits: 6

Contact hours: 72

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

• 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

• 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

• 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

• Midterm test
• Final examination
• Hometasks

Interim Assessment

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

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