Bachelor
2023/2024
Introduction to Differential Geometry
Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'
Category 'Best Course for New Knowledge and Skills'
Type:
Elective course (Applied Mathematics and Information Science)
Area of studies:
Applied Mathematics and Information Science
Delivered by:
Big Data and Information Retrieval School
Where:
Faculty of Computer Science
When:
4 year, 3 module
Mode of studies:
distance learning
Online hours:
32
Open to:
students of one campus
Instructors:
Dmitry Trushin
Language:
English
ECTS credits:
5
Contact hours:
40
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.
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.
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.