Class timings: Tue Thu 3:30-5pm First class is on Thu, Jan 1
Course description: https://math.iisc.ac.in/all-courses/ma235.html
Course TA: Ritvik Saharan (ritviks@), Office hour: Thu 11:30-12:30 at L21
Office hours: Wed 4-5pm or by appointment
Recommended books:
John Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, Springer-Verlag 2012
Dennis Barden and Charles Thomas, An Introduction to Differential Manifolds, World Scientific 2003.
Michael Spivak, Comprehensive Introduction to Differential Geometry, Vol 1, Publish or Perish, 2005.
Here is a Quanta article about the subject: https://www.quantamagazine.org/what-is-a-manifold-20251103
Grading policy: 20% Quizzes, 40% Midterm exam , 40% Final exam.
Week 1 Definitions of smooth manifolds, examples, existence of a partition of unity.
Week 2 Review of multivariable calculus, Inverse function theorem, Definition of submanifolds and examples.
Week 3 Submersions, immersions, local submersion/immersion theorem, examples of Lie groups.
Week 4 More on submersions: local section theorem, smooth covering maps, embeddings, compact manifold as a submanifold.
Week 5 Derivations, definition of the tangent space of a manifold at a point, Derivative of smooth maps between manifolds, the tangent bundle.
Week 6 Sard's theorem, regular and critical values, measure zero sets, mini-Sard theorem. Proof of the weak Whitney embedding theorem, for both compact and non-compact manifolds.
Week 7 Manifolds with boundary, No retraction theorem, Vector fields: definition as a section of the tangent bundle, and as a derivation, examples: pushforward via diffeomorphisms, left-invariant vector fields on a Lie group.
There will be weekly quizzes based on the homework.
Homework 1 -- Quiz on 15th January.
Homework 2 -- Quiz on 22nd January.
Homework 3 -- Quiz on 29th January.
Homework 4 -- Quiz on 5th February.
Homework 5 -- to be discussed on 12th Feb, more in Ritvik's review session on Saturday 14th Jan 3-5pm at LH-2