Class timings: Tue Thu 3:30-5pm
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.
Week 8 Review, Midterm week, discussion of exam.
Week 9 Lie bracket of vector fields, examples, Lie algebra, Flow of a vector field, Lie derivative, Commuting flows, statement of the Frobenius theorem.
Week 10 Covector fields i.e. differential 1-forms, pullback, Line integrals, exact 1-forms. (Covariant) tensors and tensor fields, symmetric tensors (eg. Riemannian metrics), alternating tensors.
Week 11 Alternating k-tensors on an n-dimensional vector space, examples, wedge product and its properties , elementary alternating k-tensors, differential k-forms as alternating k-tensor fields, pullback of k-forms.
Week 12 Integration of a compactly supported n-form on an n-dimensional oriented manifold, definition of oriented manifold and examples notion of orientation on a vector space, equivalent definition of an oriented manifold in terms of a non-vanishing n-form, real projective plane is non-orientable, an oriented surface in R^3 has a nowhere-tangent vector field.
Week 13 A level set of a function is oriented, induced orientation on boundary of a manifold. Exterior derivatives: definition in R^n, definition on manifolds, properties including commuting with pullbacks, and d^2=0; the notion of closed forms and exact forms.
Week 14 Back to integrals: some properties, including the integral of the orientation form is positive. Stokes theorem and its proof. Some applications, including the No Retraction theorem (for compact oriented manifolds) and proof of the Hairy Ball Theorem. Definition of de Rham cohomology groups (examples: S^1 and R^2); their functorial properties. Notion of degree of a self-map of a sphere.
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
Homework 6 -- Quiz on 12th March
Homework 7 -- Quiz on 26th March
Homework 8 -- Quiz on 9th April (2pm)
Homework 9 -- to be discussed in the review session (Saturday, April 18, 11-12 at LH-4)
Final exam -- on 29th April 2-5pm at LH-4
Graded final exams can be seen on Monday, May 4th and May 5th from 4-5pm at my office.