Details of MA4203 (Spring 2013)
| Level: 4 | Type: Theory | Credits: 3.0 |
| Course Code | Course Name | Instructor(s) |
|---|---|---|
| MA4203 | Differential Geometry | Veerendra Vikram Awasthi |
| Syllabus |
|---|
| Smooth Manifolds: Topological manifolds, smooth structures and local coordinates, examples.
Smooth Maps: Smooth functions, partitions of unity. Tangent Bundle: Tangent vectors, tangent spaces, computation in local coordinates, tangent bundle, tangent of mappings, cotangent bundle; introduction to vector bundle. Submanifolds: Submersions, immersions and embeddings; submanifolds; inverse function theorem. Vector Fields: Vector fields and integral curves, flows, fundamental theorem on flows, complete vector fields; Lie derivative and Lie bracket, basic properties. Tensors: Algebra of tensors, tensors and tensor fields on manifolds, symmetric tensors, Riemannian metrics. Differential Forms: Exterior algebra, differential forms on manifolds, exterior derivatives; closed and exact forms, Poincare lemma; symplectic forms, Darboux theorem. Integration on Manifolds: Orientations, integration of differential forms, Stokes' theorem. |
| References |
|---|
| 1. Conlon, L., Differentiable Manifolds, Birkhuser-Verlag, 2008.
2. Kumaresan, S., A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002. 3. Lee, J. M., Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer-Verlag, 2002. 4. Milnor, J, W., Topology from the Differentiable Viewpoint, Princeton University Press, 1997. 5. Mukherjee, A., Topics in Differential Topology, Hindustan Book Agency, 2005. 6. Tu, L. W., An Introduction to Manifolds, Universitext, Springer-Verlag, 2007. |
Course Credit Options
| Sl. No. | Programme | Semester No | Course Choice |
|---|---|---|---|
| 1 | IP | 2 | Core |
| 2 | IP | 4 | Not Allowed |
| 3 | MS | 8 | Core |
| 4 | RS | 1 | Elective |