Details of MA4203 (Spring 2015)

Level: 8 Type: Theory Credits: 3.0

Course CodeCourse NameInstructor(s)
MA4203 Differential Geometry Saugata Bandyopadhyay

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. L. Conlon, Differentiable Manifolds, Birkhuser-Verlag, 2008.

  2. S. Kumaresan, A Course in Differential Geometry and Lie Groups, Hindustan Book Agency, 2002.

  3. J. M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer-Verlag, 2002.

  4. J. W. Milnor, Topology from the Differentiable Viewpoint, Princeton University Press, 1997.

  5. A. Mukherjee, Topics in Differential Topology, Hindustan Book Agency, 2005.

  6. L. W. Tu, An Introduction to Manifolds, Universitext, Springer-Verlag, 2007.

Course Credit Options

Sl. No.ProgrammeSemester NoCourse Choice
1 IP 2 Core
2 IP 4 Not Allowed
3 MR 2 Not Allowed
4 MR 4 Not Allowed
5 MS 8 Core
6 RS 1 Not Allowed
7 RS 2 Not Allowed