Differential Topology

Code: MA 644 | L-T-P-C: 4-0-0-8

Prerequisites: MA549 Topology or equivalent

Preamble: This course is an introduction to the modern methods of studying calculus on manifolds. It covers topics like the derivative of a function between manifolds, performing integration on a manifold, Stokes' Theor em as a generalization of the Fundamental Theorem of Calculus and ends with the remarkable theorem of de Rham - which establishes an intricate link between objects on a manifold derived from the disparate sources of topology and calculus.

Syllabus: Differential manifolds, Smooth maps, Tangent spaces, Tangent bundle, Differential of a smooth map. Submersions, Immersions, Embeddings, Submanifolds and Sard's Theorem. Cotangent Bundle, Tensors, Differential forms, Exterior derivative, Closed and exact forms, Poincare Lemma. Orientation on manifolds, Integration, Stokes' Theorem, de Rham cohomology and the theorem of de Rham.

Texts/References:

1. V. Guillemin and A. Pollack, Differential Topology, American Mathematical Society, Reprint Edition, 2010
2. J. M. Lee, Introduction to Smooth Manifolds, Springer, 2002.
3. M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol.1, Publish or Perish, 3rd Edn., 1999