Course

Manifolds (MAT510)

An introduction to smooth manifolds and related concepts in differential geometry.


Dette er emnebeskrivelsen for studieåret 2021-2022

See course description and exam/assesment information for this semester (2024-2025)

Semesters

Fakta

Emnekode

MAT510

Vekting (stp)

10

Semester undervisningsstart

Autumn

Undervisningsspråk

English

Antall semestre

1

Vurderingssemester

Autumn

Timeplan

Vis timeplan

Litteratur

Pensumlisten finner du i Leganto

Content

This course gives an introduction to smooth manifolds and related concepts in differential geometry. Smooth functions, directional derivatives, tangent vectors and differential forms on a Euclidean space will be reviewed. The basics of point-set topology will be introduced and the concept of a topological manifold will be defined. The notion of a smooth manifold will be introduced, with a plethora of familiar (and perhaps not so familiar) examples. Smooth maps between manifolds will be defined, together with the important concept of a diffeomorphism. Further concepts like tangent spaces, differentials, submanifolds, vector fields and integral curves will also be developed.   

Learning outcome

After completing this course, the student should understand how familiar concepts from vector calculus are subsumed by the framework of differential forms on a Euclidean space. The student should also have a decent grasp of the key notions in point-set topology, and be able to state definitions and give examples. Furthermore, the student should be able to perform simple calculations on smooth manifolds and work out detailed properties in examples.

Forkunnskapskrav

Ingen

Anbefalte forkunnskaper

Mathematical Methods 1 (MAT100), Linear Algebra (MAT110), Real and Complex Calculus (MAT210), Abstract Algebra (MAT250), Vector Analysis (MAT300), Differential Equations (MAT320)

Exam

Fagperson(er)

Head of Department:

Bjørn Henrik Auestad

Course coordinator:

Paul Francis de Medeiros

Method of work

5-6 hours lecturing and problem solving per week. Language of tuition: English.

Overlapping

Emne Reduksjon (SP)
Mathematical Modelling (MAT500_1) , Manifolds (MAT510_1) 10

Åpent for

Admission to Single Courses at the Faculty of Science and Technology
Mathematics and Physics - Master of Science Degree Programme Mathematics and Physics - Five Year Integrated Master's Degree Programme

Emneevaluering

Use of evaluation forms and/or conversation for students' evaluation of the course and teaching, according to current guidelines.

Litteratur

Book

An Introduction to Manifolds [electronic resource] Tu, Loring W., New York, NY :, Springer New York ; Imprint Springer, 1 online resource (426 p.), 2011., isbn:1-4419-7400-8, https://bibsys-ur.userservices.exlibrisgroup.com/view/uresolver/47BIBSYS_UBIS/openurl-UIS?ctx_enc=info:ofi/enc:UTF-8&ctx_id=10_1&ctx_tim=2021-05-04T12:41:43IST&ctx_ver=Z39.88-2004&url_ctx_fmt=info:ofi/fmt:kev:mtx:ctx&url_ver=Z39.88-2004&rfr_id=info:sid/primo.exlibrisgroup.com-BIBSYS_ILS&req_id=&rft_dat=ie=47BIBSYS_UBIS:5135272440002208,ie=47BIBSYS_UBB:51111880960002207,ie=47BIBSYS_UBA:5137704090002209,ie=47BIBSYS_UBTO:5184179950002205,ie=47BIBSYS_UBO:51175608100002204,ie=47BIBSYS_NTNU_UB:51167289320002203,ie=47BIBSYS_NMBU:5117714770002213,ie=47BIBSYS_SIRUS:5135411620002256,ie=47BIBSYS_AHUS:5119914280002263,ie=47BIBSYS_NETWORK:71523580190002201,language=eng,view=UBIS&svc_dat=viewit&u.ignore_date_coverage=true&user_ip=10.16.56.55&req.skin=primoView online

Book

Introduction to Topological Manifolds [electronic resource] Lee, John., New York, NY :, Springer New York ; Imprint Springer, 1 online resource (444 p.), 202, 2011., isbn:1-4419-7940-9, https://bibsys-ur.userservices.exlibrisgroup.com/view/uresolver/47BIBSYS_UBIS/openurl-UIS?ctx_enc=info:ofi/enc:UTF-8&ctx_id=10_1&ctx_tim=2021-05-04T12:41:43IST&ctx_ver=Z39.88-2004&url_ctx_fmt=info:ofi/fmt:kev:mtx:ctx&url_ver=Z39.88-2004&rfr_id=info:sid/primo.exlibrisgroup.com-BIBSYS_ILS&req_id=&rft_dat=ie=47BIBSYS_UBIS:5136359430002208,ie=47BIBSYS_UBB:51109830870002207,ie=47BIBSYS_UBA:5136512850002209,ie=47BIBSYS_UBTO:5176379690002205,ie=47BIBSYS_UBO:51173773660002204,ie=47BIBSYS_NTNU_UB:51166482420002203,ie=47BIBSYS_NMBU:5117957260002213,ie=47BIBSYS_SIRUS:5134076130002256,ie=47BIBSYS_AHUS:5119933830002263,ie=47BIBSYS_NETWORK:71524453880002201,language=eng,view=UBIS&svc_dat=viewit&u.ignore_date_coverage=true&user_ip=10.16.56.55&req.skin=primoView online

Book

Differential topology Guillemin, Victor,, Pollack, Alan,, Providence, Rhode Island :, AMS Chelsea Publishing, 1 online resource (242 pages) :, 2014., isbn:1-4704-1135-0,

Kontakt

Head of Department:

Bjørn Henrik Auestad

Course coordinator:

Paul Francis de Medeiros

Tilbys av

Faculty of Science and Technology

Department of Mathematics and Physics

The course description is retrieved from FS (Felles studentsystem). Version 1