MAT 338. Differentiable manifolds and Lie groups - Autumn 2017 - Vamsi Pingali

The text we will (largely) be following is ``A comprehensive introduction to differential geometry (Vol 1)" by Spivak. We will try to cover as much of this book as possible. Here are some other references :
1. Kumaresan, S., A Course in Differential Geometry and Lie Groups
2. Warner, F., Foundations of Differentiable Manifolds and Lie Groups
3. John Lee, Introduction to smooth manifolds.
4. Milnor, Topology from a differentiable viewpoint.
5. T. Frankel, Geometry and physics.
6. Hawking and Ellis, The large scale structure of spacetime.
7. Loring Tu, An introduction to manifolds.

Instructor : Vamsi Pritham Pingali,

Office : N23 in the mathematics building. Office hours : Wed from 3-4 pm. (Feel free to come during other times after emailing me first.)

Classroom and timings : Monday, Wednesday, Friday from 2-3 in LH-5.

The Grading policy : 20% for Homeworks, Midterm-30%, and 50% for the Final. Under NO circumstances will makeup exams be held for the midterm. If you have a valid and provable excuse, (Schedule conflicts with other courses do NOT constitute as valid excuses. You are supposed to resolve them before registering for the courses.) then your performance on the other exams shall determine your grade on your midterms.

Exams :

The Midterm shall be held on September 22 (Friday) from 2-3:30 in the class. (Syllabus : Everything we did up to (and including) Frobenius' theorem.)

The final for this course will be held on December 7 (Thursday) from 9:00-12:00 in our usual classroom. (Syllabus : Everything we did except Riemannian geometry.)

Ethics: Read the information on the IISc student ethics page. In short, cheating is a silly thing. Don't do it. As for homeworks, write them up on your own. You are allowed to discuss them amongst yourselves but please write the solutions on your own. That said I must hasten to add that you learn mathematics best when you solve the problems entirely by yourself.

Here is the tentative schedule. (It is subject to changes and hence visiting this webpage regularly is one of the best ideas in the history of best ideas.)



 Syllabus to be covered


31 July - 6 Aug

Logistics, history and motivation, definition of a topological manifold (Wednesday notes), Friday is a holiday


7 Aug - 13 Aug

Definition of a smooth manifold, examples (Monday notes), Maps between manifolds (Wednesday notes,Friday notes),


14 Aug - 20 Aug

Inverse and implicit function theorems(Wednesday notes), Partitions-of-unity and the Whitney embedding theorem (Friday notes)


21 Aug - 27 Aug

Sard's theorem (Monday notes), Cotangent bundle, Tangent bundle, etc (Wednesday notes)


28 Aug - 3 Sept

Cotangent bundle, Tangent bundle, etc (Monday notes), (Wednesday notes), (Friday notes)


4 Sept - 10 Sept

Vector fields and flows (Monday notes, Wednesday notes), Lie derivative ((Friday notes))


11 Sept - 16 Sept

Lie derivative (Monday notes), local Frobenius theorem (Wednesday notes), Lie groups ((Friday notes))


17 Sept - 23 Sept

No classes but the midterm is from 2-3:30 in LH-5 (the usual classroom) on Friday Sept 22nd.


24 Sept - 30 Sept

Lie groups (Monday notes), Revision (Wednesday notes)


1 Oct - 6 Oct

Revision (Wednesday notes), Lie algebras (Friday notes)


7 Oct - 13 Oct

Lie algebras (Wednesday notes), (Friday notes)


14 Oct - 20 Oct

Tensors (Monday notes), (Friday notes)


21 Oct - 27 Oct

Alternating tensors (Monday notes), Orientation (Wednesday notes), Differential forms (Friday notes)


28 Oct - 3 Nov

The exterior derivative (Monday notes), Closed and exact forms, Poincare's lemma (Wednesday notes), Integration (Friday notes)


4 Nov - 10 Nov

Stokes' theorem (Monday notes), De Rham cohomology (Wednesday notes), (Friday notes)


11 Nov - 17 Nov

De Rham cohomology (Monday notes), Degree of a proper map ((Wednesday notes)), ((Friday notes))


18 Nov - 24 Nov

Definition of a metric on a vector bundle (Monday notes), Examples of Riemannian metrics and volume forms (Wednesday notes), Geodesic equation (Friday notes)

You sweet summer children, here is the winter HW...

A big thank you to Kesav, Raghavendra, Sayantan, Sarvesh, and Chaitanya (and whoever else in the future) for latexing their HW. I shall post the ``best written" solutions (whilst removing responsibility from the said students for typographical errors) on the webpage so that others may benefit.


 To be handed to me on

 Homework (subject to changes; please check regularly)


4th Aug

No HW due


16th Aug (Wednesday)

HW 1


23rd Aug (Wednesday)

HW 2


1st Sept (Friday)

HW 3 (Thanks Sarvesh for pointing out something that makes the question a little less confusing. )


8th Sept (Friday)

HW 4


15th Sept (Friday)

HW 5


6th Oct (Friday)

HW 6


23rd Oct (Monday)

HW 7


8th November (Wednesday)

HW 8


22nd November (Wednesday)

HW 9