MT3184
Differential geometry
and Lie groups
This lecture notes will be updated frequently. Please let me know if you find any typos. If you think some idea needs more explanation, do let me know. You can reach me on praphullakoushik16@gmail.com
Lecture timings:
Monday 9AM-9:55 AM
Tuesday 9AM-9:55 AM
Wednesday 9AM-9:55 AM
Start date of instruction:
8th August 2022
Last date of instruction:
23rd November 2022
Office Hours :
By email appointment
Teaching days
8th August
10th August
16th August
17th August
22nd August
23rd August
24th August
29th August
30th August
5th September
6th September
7th September
12th September
13th September
14th September
19th September
20th September
21st September
10th October
11th October
12th October
17th October
18th October
19th October
25th October
26th October
31st October
1st November
2nd November
7th November
9th November
14th November
15th November
16th November
21st November
22nd November
23rd November
Non teaching days:
9th August
15th August
31st August
26th September
27th September
28th September
3rd October
4th October
5th October
24th October
8th November
This page contains information about the course MT3184 Differential geometry and Lie groups that I will be teaching in the semester August-November 2022 in IISER Pune.
Prerequisites for the course:
Some understanding of topology (separation axioms, countability axioms, compactness, connectedness, quotient spaces). It would be good if you already know examples of spaces that satisfy certain property (but not the other).
Some understanding of multi variable calculus (partial derivative, total derivative of functions between subsets of R^n, smooth functions between subsets of R^n, Taylor's theorem)
Some understanding of group theory (definition of group, some basic properties), some understanding of linear algebra (vector spaces, linear maps between vector spaces, some basic properties).
There are 37 teaching days (if everything goes as per academic calendar). Out of which we will use 30 teaching days for the below mentioned lecture sessions and 4 teaching days for problem sessions. The remaining 3 teaching days will be reserved for discussion of some special topics (based on interest of students).
Ambitious/tentative list of topics to be covered (one topic per session):
(Introduction to the course and) Review of multivariable calculus.
Review of multilinear algebra
Definition of smooth manifolds (and some examples).
Definition of smooth map between smooth manifolds (and some examples).
Smooth "bump" functions and partition of unity.
Tangent space at a point on smooth manifold (some properties, some computations).
smooth map between smooth manifolds inducing linear map between tangent spaces.
Two classes of smooth maps : immersions and submersions.
Two classes of submanifolds: immersed submanifolds, embedded submanifolds.
Whitney embedding theorem (at least for compact manifolds).
Tangent bundle construction (some idea about general notion of vector bundle).
vector fields on smooth manifolds.
Lie bracket of vector fields.
integral curves (and flows) associated to vector fields.
Cotangent bundle construction.
differential forms on smooth manifolds.
Lie derivative, interior derivative, and exterior derivative operators.
Cartan's magic formula (relating Lie derivative, interior derivative, exterior derivative).
Orientation on a smooth manifold.
integration on oriented smooth manifolds.
de Rham cohomology and de Rham theorem (assuming the student already saw some cohomology).
Stoke's theorem in the set up of differential forms.
Definition of Lie groups (and some examples).
Lie algebra of a Lie group (some computations)
Exponential map in the setup of Lie groups and Lie algebras.
The Baker-Campbell-Hausdorff formula (really ambitious topic).
Lie group action on a smooth manifold, and the quotient manifold theorem.
(universal) covering space of a smooth manifold.
Foliations and distributions.
Frobenius theorem on Integrable distributions.
Note : It is yet to be decided if the topics mentioned in red color are to be discussed in class or to be left to students for further reading.
Update: The topics mentioned in red color was not discussed due to time constraints.
Special topics (to be covered based on interest/feedback of students)
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Problem sessions (viva session):
We will have 4 problem/viva sessions, one each on the following days:
30th August
21st September
31st October
23rd November
Grading :
There are three components of grading
Viva/Assignments (to be happened in problem sessions; assignments),
Mid semester exam (date is TBA),
End semester exam (date is TBA)
Your final grade will be the maximum of the following two scores
20% viva + 40% mid sem + 40% end sem
50% mid sem + 50% mid sem.
Exercises :
In the lecture session, I will leave some small steps as exercises. You are strongly encouraged to fill those gaps.
Assignments:
Mid semester examination:
MT 3184 mid semester examination question paper
MT 3184 mid semester examination solutions
End semester examination:
Main references:
An Introduction to Manifolds by Loring W. Tu
Introduction to Smooth Manifolds by John M. Lee
Suggested further reading (matheamtics oriented)
Foundations of Differentiable Manifolds and Lie Groups by Frank W. Warner
Differential Forms and Applications by Manfredo P. do Carmo
A Geometric Approach to Differential Forms by David Bachman
Sections 1.1 to 1.4 of Foundations of Differential Geometry, Volume 1 by Kobayashi and Nomizu
Geometry of Differential Forms by Shigeyuki Morita.
Suggested futher reading (physics oriented)
Modern Differential Geometry for Physicists by Chris J. Isham
Chapters 5 and 6 of Geometry, topology and physics by Mikio Nakahara
Some youtube videos on differential geometry and/or Lie groups:
Lecture 1 to Lecture 6 in this YouTube playlist delivered by Frederic P Schuller are relevant for our course. Before we start the course, you should have nodding familiarity with contents in lecture 1 of this playlist.
Lecture 1 to Lecture 13 in this YouTube playlist delivered by Frederic P Schuller are relevant for our course.
All lectures in this YouTube playlist delivered by Sunil Mukhi are relevant for our course.
Goal :
By the end of the course, you should have enough background to read the following topics,
Riemannian geometry (starting from Introduction to Riemannian Manifolds by John M. Lee
Symplectic geometry (starting from chapter 22 of Introduction to Smooth Manifolds by John M. Lee)
Poisson geometry (starting from Lectures on Poisson Geometry by Crainic, Fernandes, and Marcut).
Complex geometry (starting from Complex Geometry by Daniel Huybrechts
vector bundles, principal bundles (starting from Foundations of Differential Geometry, Volume 1 and Volume 2 by Kobayashi and Nomizu).