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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

• 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 #### Interim Assessment

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