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Бакалавриат 2020/2021

Введение в дифференциальную геометрию

Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Лучший по критерию «Новизна полученных знаний»
Статус: Курс обязательный (Прикладная математика и информатика)
Направление: 01.03.02. Прикладная математика и информатика
Когда читается: 4-й курс, 3 модуль
Формат изучения: с онлайн-курсом
Язык: английский
Кредиты: 5
Контактные часы: 12

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

Learning Objectives

  • To know the basics of differential geometry
Expected Learning Outcomes

Expected Learning Outcomes

  • To know the theory of differentiation and integration on manifolds
  • To know the basic concepts of differential geometry
Course Contents

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

Assessment Elements

  • non-blocking Current assessment
  • non-blocking Exam
Interim Assessment

Interim Assessment

  • Interim assessment (3 module)
    0.5 * Current assessment + 0.5 * Exam
Bibliography

Bibliography

Recommended Core Bibliography

  • Кузовлев В.П., Подаева Н.Г. - Курс геометрии: элементы топологии, дифференциальная геометрия, основания геометрии - Издательство "Физматлит" - 2012 - 208с. - ISBN: 978-5-9221-1360-1 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/59618

Recommended Additional Bibliography

  • Ильин В.А., Позняк Э.Г. - Линейная алгебра. - Издательство "Физматлит" - 2007 - 280с. - ISBN: 978-5-9221-0481-4 - Текст электронный // ЭБС ЛАНЬ - URL: https://e.lanbook.com/book/2178