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 R^n, 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