Introduction to Differential Geometry

2023/2024

Учебный год

ENG

Обучение ведется на английском языке

5

Кредиты

Кто читает:

Департамент больших данных и информационного поиска

Статус:

Курс по выбору

Когда читается:

4-й курс, 3 модуль

Преподаватель

Трушин Дмитрий Витальевич

Course Syllabus

Abstract

Differential geometry is mathematical analysis together with differential equations and linear algebra together with optimization theory. It has always developed under the great influence of physics and has always found applications both in applied sciences and within the abstract areas of mathematics. The course will cover the most basic things. It will outline what a smooth manifold is and how the mappings between them are arranged. Manifolds are nonlinear surfaces of arbitrary fixed dimensions, generalizations of linear spaces. Students will learn how to properly differentiate and integrate on manifolds. Differentiation will lead to covariant derivatives, and integration to the theory of differential forms and de Rham cohomology.

Learning Objectives

• To know the basics of differential geometry

Expected Learning Outcomes

• know the basic concepts of differential geometry
• know the theory of differentiation and integration on manifold

Course Contents

• Topology, topological manifolds, morphisms of manifolds, operations on manifolds
• Smooth structure, atlas, smooth manifolds, morphisms of smooth manifolds (structure results), submanifolds.
• Tangent and cotangent spaces, vector bundles, tangent, cotangent, and tensor bundles.
• Covariant derivative (or connection), Christoffel symbols, metric, Riemannian connection, parallel transporta- tion, geodesics.
• Differential forms, orientation. Integration of differential forms. Complex of differential forms and de Rham cohomology.

Assessment Elements

• Current assessment
• Exam

Interim Assessment

• 2023/2024 3rd module
  0.5 * Current assessment + 0.5 * Exam

Bibliography

Recommended Core Bibliography

• Кузовлев, В. П. Курс геометрии: элементы топологии, дифференциальная геометрия, основания геометрии : учебник / В. П. Кузовлев, Н. Г. Подаева. — Москва : ФИЗМАТЛИТ, 2012. — 208 с. — ISBN 978-5-9221-1360-1. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/59618 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.

Recommended Additional Bibliography

• Ильин, В. А. Линейная алгебра : учебник / В. А. Ильин, Э. Г. Позняк. — 6-е изд., стер. — Москва : ФИЗМАТЛИТ, 2020. — 280 с. — ISBN 978-5-9221-0481-4. — Текст : электронный // Лань : электронно-библиотечная система. — URL: https://e.lanbook.com/book/185610 (дата обращения: 00.00.0000). — Режим доступа: для авториз. пользователей.

Authors

• TRUSHIN DMITRIY VITALEVICH