Differentiable Manifolds 1
By Mehdi Nadjafikhah (IUST)

References:

        J.M. Lee, Introduction to topological manifolds, GTM, Springer, 2000.

        J.M. Lee, Introduction to smooth manifolds, GTM, Springer, 2013.

        M. Nadjafikhah, An introduction to differentiable manifolds, IUST, 2017.

        M. Spivak, Calculus on manifolds: A modern approach to classical theorems of advanced calculus, 1965. Persian translated by M. Nadjafikhah, 2004 (pdf)

        F.W. Warner, Foundations of differentiable manifolds and Lie groups, (Translated in to Persian by M. Nadjafikhah) (pdf)

        G. Rudolph, M. Schmidt, Differential geometry and mathematical physics, Part I: Manifolds, Lie groups and Hamiltonian systems, Springer, 2013.

        L.Tu, An introduction to smooth manifolds, Springer Verlag, 2012 (Translated in to M. Nadjafikhah and A.R. Forough) (pdf)



Course materials:

 

lecture

title

contents

video

1

‎introduction

history and motivations

V01

2

topological manifold

topology‎, ‎ddefinition of topological space‎, metric‎, ‎topological manifold‎, ‎atlas

V02

3

‎quotient manifold‎

‎‎‎quotient topology‎, ‎equivalence relation on topological space‎s, ‎real projective space, quotient manifold‎

V03

4

smooth manifold

reminder‎ ‎differentiable function‎, smooth atlas‎, maximal atlas‎, ‎smooth manifold‎

V04

5

construction on new manifolds

multiplication manifold‎, ‎a simpler way to definition of a smooth structure‎

V05

6

smooth map‎s

definition and properties of smooth map‎s

V06

7

‎regular and smooth map‎s

algebraic properties of smooth map sets‎, regular map‎s, rank‎, regular point‎s

V07

8

regular maps

treatment of regular maps‎, ‎inverse function theorem‎, ‎constant rank theorem‎, ‎immersion‎, ‎submersion‎, local diffeomorphism

V08

9

sub-manifold I

immersed sub-manifold‎s

V09

10

sub-manifold II

regular sub-manifold‎, regular value‎

V10

11

sub-manifold III

‎‎restriction of range of functions, regular sub-manifold‎s, constant-rank level set theorem‎, level set theorem‎

V11

12

tangent vectors

directional derivative‎, generalization of tangent vector to smooth manifolds‎,

tangent space to Rn, germs of functions, derivative functions on the manifolds‎, tangent space to manifolds

V12

13

tangent bundle

reminder of lecture 12‎, change the coordinates for the derivative‎, tangent bundle

V13

14

differential map

differential map‎, properties of differential‎, ‎curve‎, physical interpretation of tangent space‎

V14

15

vector fields

vector field as derivative operator‎, ‎a few criterion for the smooth vector field‎, ‎Lie derivative of vector field‎, ‎Lie bracket‎, ‎coordinate representation for Lie bracket‎, properties of Lie bracket‎, related vector fields‎

V15

16

‎‎‎related vector fields‎

f-related vector fields‎, the push-forward of vector fields‎, tangent space on sub-manifold‎s

V16

17

integral curve‎s

integral curve‎s, maximal integral curve‎s, examples

V17

18

flow

flows of vector fields‎, infinitesimal generator‎ of a low

V18

19

Lie derivative

Lie derivative‎, rectifying theorem‎, characteristic method for solving pseudo-linear equations‎

V19

20

tensor calculations

tensor product‎, basis for tensors‎

V20

21

‎‎types of tensors‎

reminder‎ permutation‎, ‎symmetry and Alternative tensors‎, ‎contraction‎,

‎symmetric and alternative tensors‎, ‎wedge and symmetry product‎, ‎Taylor series‎

V21

22

 

 

V22

23

application of multi-vector‎s

exterior algebra of vector space‎, inner product‎, ‎direction in vector space‎, ‎volume unit in vector space‎

V23

24

vector bundles

push-forward and pullback‎, vector bundles, ‎smooth section‎

V24

25

‎smooth sections

‎smooth sections‎, simple definition of vector bundle‎, smooth frame‎, criterion of smooth of sections‎

V25

26

 

 

V26

27

Poincare theorem

exterior differential, exact and closed forms, Poincare theorem

V27

 

Telegram channel of this course

Some Exams: Mid-term, Final

Last modified date: August 16, 2017