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, vamsipingali@math.iisc.ernet.in.
Office : N23 in the mathematics building. Office hours : Wed from 34 pm. (Feel free to come during other times after emailing me first.)
Classroom and timings : Monday, Wednesday, Friday from 23 in LH5.
The Grading policy : 20% for Homeworks, Midterm30%, 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 23:30 in the class. (Syllabus : Everything we did up to (and including) Frobenius' theorem.)
The final for this course will be held on TBA.
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.)
Wk 
Dates 
Syllabus to be covered 
1 
31 July  6 Aug 
Logistics, history and motivation, definition of a topological manifold (Wednesday notes), Friday is a holiday 
2 
7 Aug  13 Aug 
Definition of a smooth manifold, examples (Monday notes), Maps between manifolds (Wednesday notes,Friday notes), 
3 
14 Aug  20 Aug 
Inverse and implicit function theorems(Wednesday notes), Partitionsofunity and the Whitney embedding theorem (Friday notes) 
4 
21 Aug  27 Aug 
Sard's theorem (Monday notes), Cotangent bundle, Tangent bundle, etc (Wednesday notes) 
5 
28 Aug  3 Sept 
Cotangent bundle, Tangent bundle, etc (Monday notes), (Wednesday notes), (Friday notes) 
6 
4 Sept  10 Sept 
Vector fields and flows (Monday notes, Wednesday notes), Lie derivative ((Friday notes)) 
7 
11 Sept  16 Sept 
Lie derivative (Monday notes), local Frobenius theorem (Wednesday notes), Lie groups ((Friday notes)) 
8 
17 Sept  23 Sept 
No classes but the midterm is from 23:30 in LH5 (the usual classroom) on Friday Sept 22nd. (Midterm solutions) 
9 
24 Sept  30 Sept 
Lie groups (Monday notes), Revision (Wednesday notes) 
10 
1 Oct  6 Oct 
Revision (Wednesday notes), Lie algebras (Friday notes) 
11 
7 Oct  13 Oct 
Lie algebras (Wednesday notes), (Friday notes) 
12 
14 Oct  20 Oct 
Tensors (Monday notes), (Friday notes) 
13 
21 Oct  27 Oct 
Alternating tensors (Monday notes), Orientation (Wednesday notes), Differential forms (Friday notes) 
14 
28 Oct  3 Nov 
The exterior derivative (Monday notes), Closed and exact forms, Poincare's lemma (Wednesday notes), Integration (Friday notes) 
15 
4 Nov  10 Nov 
Stokes' theorem (Monday notes), De Rham cohomology (Wednesday notes), (Friday notes) 
16 
11 Nov  17 Nov 
De Rham cohomology (Monday notes), Degree of a proper map ((Wednesday notes)), ((Friday notes)) 
17 
18 Nov  24 Nov 
Definition of a metric on a vector bundle (Monday notes), Examples of Riemannian metrics and volume forms (Wednesday notes) 
Wk 
To be handed to me on 
Homework (subject to changes; please check regularly) 
1 
4th Aug 
No HW due 
3 
16th Aug (Wednesday) 

4 
23rd Aug (Wednesday) 

5 
1st Sept (Friday) 
HW 3 (Thanks Sarvesh for pointing out something that makes the question a little less confusing. Solutions)) 
6 
8th Sept (Friday) 

7 
15th Sept (Friday) 

8 
6th Oct (Friday) 

11 
23rd Oct (Monday) 

13 
8th November (Wednesday) 

15 
22nd November (Wednesday) 