Class timings: MWF 3-4
Recommended books: F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer GTM 1983.
J.M. Lee, Introduction to smooth manifolds, Springer GTM 2012.
S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.
V. Guillemin and A. Pollack, Differential Topology, AMS 1974, reprint edition 2010.
Grading policy: 50% Homework , 20% Midterm exam, 30% Final exam.
The final exam is due at 5pm on June 7th. No late submissions will be allowed.
There will be no lectures during the midterm week, April 12-16.
There will be a midterm exam for those crediting the course. Details will be announced over email.
There will be no class on Friday, March 19th. We will make up for it by extending the lecture hours on a couple of Mondays.
We decided to change the class timings to MWF 3-4. The next meeting is on Friday, 5th March, from 3-4pm.
The first meeting is on Tuesday, March 2 from 3:30-5pm. We shall discuss and finalize any change in class timings before we start with the course material.
The MS Teams code is now up on the IISc Intranet page . Non-IISc students should email me, and I'll add you as a "guest" to the Team.
Classes will be online, at the beginning over Microsoft Teams. The first meeting link will be put up here, and on the IISc Intranet Bulletin board, before the meeting time.
Weekly assignments will be posted on the Teams channel, and the problems will be available here.
The recorded videos will be available on the Teams channel.
Lecture 1 Basic definitions and examples
Lecture 2 Review of first lecture, existence of a partition of unity
Lecture 3 Review of multivariable calculus, the Inverse Function Theorem
Lecture 4 Manifolds via the IFT, notion of a submanifold, first examples of Lie groups
Lecture 5 Tangent vectors, derivations, and tangent spaces on manifolds
Lecture 6 More on tangent spaces and derivatives, submersions and the Preimage Theorem
Lecture 7 Immersions and embeddings
Lecture 8 Embedding a compact manifold into some Euclidean space, Sard's theorem
Lecture 9 Whitney embedding theorem (compact case)
Lecture 10 Whitney embedding (non-compact case)
Lecture 11 Manifolds with boundary, proof of the No Retraction theorem
Lecture 12 Vector fields, left-invariant vector fields on Lie groups, the Lie bracket
Lecture 13 More on the Lie bracket, Lie algebras of Lie groups
Lecture 14 Flows of vector fields
Lecture 15 The Lie bracket
Lecture 16 Discussion of midterm exam, Commuting flows and the Lie bracket
Lecture 17 More on commuting vector fields, the Frobenius theorem
Lecture 18 Proof of the Frobenius Theorem
Lecture 19 Global version of Frobenius Theorem
Lecture 20 Correspondence of Lie subgroups and Lie subalgebras, Vector bundles, Warmup to tensor fields
Lecture 21 The cotangent bundle, differential 1-forms, tensors
Lecture 22 Covariant tensor fields, symmetric and alternating tensors
Lecture 23 Wedge product of alternating tensors, Differential k-forms
Lecture 24 Exterior derivatives
Lecture 25 Integration of forms, orientation
Lecture 26 Integration on a manifold, induced orientation on boundary
Lecture 27 More on boundary orientation, Stokes' Theorem and its proof
Lecture 28 Finishing the proof, applications of Stokes' Theorem, de Rham cohomology
Lecture 29 More on de Rham cohomology, then back to Lie groups and Lie algebras: the exponential map
Lecture 30 More on the exponential map, application: parallel parking a car
Lecture 31 Correspondence between simply-connected Lie groups and finite dimensional Lie algebras