M483/MA876 - Introduction to Manifolds (Even Sem 2024-25)
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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, Sub-manifolds, 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 co-homology.
Prerequisites: Several variable calculus, Basic point set topology.
Texts:
Evaluation: Quizzes 40%, Midsem 30%, Final 30%.
Exams: Midsem 10-12 hrs, 3 March (Monday) 2025. Final 10-13 hrs, 8 May (Thursday) 2025.
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.
- 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.
Evaluation: Quizzes 40%, Midsem 30%, Final 30%.
Exams: Midsem 10-12 hrs, 3 March (Monday) 2025. Final 10-13 hrs, 8 May (Thursday) 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. (ps12)