Differential Geometry

Credits: 4 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 621 (Analysis I) or MAT 332/432 (Geometry of Curves and Surfaces)

Overview: Differential Geometry generalizes the calculus of several variables on Euclidean spaces to `differential manifolds’. This enables the use of analysis and linear algebra to study geometry. The two highlights of this course are the study of Lie groups and of differential forms, with the latter leading to the general Stokes’ theorem in integration.

Detailed Syllabus:

Part 1: Calculus in Rⁿ

Rⁿ as a normed linear space, derivative, chain rule, mean value theorem, directional derivatives, inverse mapping theorem, implicit function theorem, immersions and submersions, integration, higher derivatives, maxima and minima, existence of solutions of ODE.

Part 2: Differential Manifolds and Lie Groups

Differential manifolds, smooth maps and diffeomorphisms, Lie groups, tangent spaces, derivatives, immersions and submersions, submanifolds, vector fields, Lie algebras, flows, exponential map, Frobenius theorem, Lie subgroups and subalgebras.

Part 3: Differential Forms and Integration

Multilinear algebra, exterior algebra, tensor fields, exterior derivative, Poincare lemma, Lie derivative, orientable manifolds, integration on manifolds, Stokes' theorem.

References:

1. An Introduction to Differentiable Manifolds and Riemannian Geometry, by William M Boothby, 2^nd edition, Academic Press.
2. A Course in Differential Geometry and Lie Groups, by S Kumaresan, TRIM Series, Hindustan Book Agency.
3. Analysis on Manifolds, by James Munkres, Addison-Wesley.
4. Calculus on Manifolds, by Michael Spivak, Addison-Wesley.