Chitrabhanu Chaudhuri

M483/MA876 - Introduction to Manifolds (Even Sem 2024-25)

Instructor: Chitrabhanu Chaudhuri
(office M-227, email: chitrabhanu [at] niser [dot] ac [dot] in)

Lectures: 11:30-12:25 Mon, 13:30-14:25 Tue, 8:30-9:25 Thu, 9:30-10:25 Fri in M3 SMS.

Office hours: By appointment.

Academic Calendar Course Resources

Content: Differentiable manifolds and maps: Definition and examples, Inverse and implicit function theorem, Submanifolds, immersions and submersions. The tangent and cotangent bundle: Vector bundles, (co)tangent bundle as a vector bundle, Vector fields, flows, Lie derivative. Differential forms and Integration: Exterior differential, closed and exact forms, Poincare' lemma, Integration on manifolds, Stokes theorem, De Rham cohomology.

Prerequisites: Several variable calculus, Basic point set topology.

Texts:

Loring W.Tu, "An Introduction to Manifolds", Universitext, 2nd edition, 2010.
John M. Lee, “Introduction to smooth manifolds”, Springer Verlag, 2nd edition, 2013.

References:

Frank Warner, “Foundations of differentiable manifolds and Lie groups”, Springer Verlag, 2nd edition, 1983.
Michael Spivak, “A comprehensive introduction to differential geometry”, Vol. 1, 3rd edition, 1999.

Other resourses: Alexey Zinger's course material (Suggested by Ritwik)

Evaluation: Quizzes 40%, Midsem 30%, Final 30%.
Exams: Midsem 10-12 hrs, 3 March (Moday) 2025.

Material covered:

Week 1: Topological and smooth manifolds. (ps1)
Week 2: Smooth maps. (notes, ps2)
Week 3: Tangent space. (ps3)
Week 4: Submanifolds, Regular Value and Constant Rank theorems. (ps4)
Week 5: Immersion, Submersion and Embedding, Tangent bundle, Vector bundles. (ps5)
Week 6: Sections, local frames of Vector bundles, Vector Fields. (ps6)
Week 7: Integral curves and flows of vector fields, complete vector fields. (ps7)
Week 8: Bump functions, Lie Bracket of Vector fields, Manifolds with boundary.
Week 9: Manfolds with Boundary, Cotangent bundle, 1-forms. (ps8)
Week 10: Differential forms. (ps9)
Week 11: Differential forms, Exterior derivative.
Week 12: Orientation, Partitions of unity. (ps10)
Week 13: Differential forms on Manifolds with boundary. (ps11)
Week 14: Integration, Stokes' theorem.

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