M563 - Differentiable Manifolds and Lie Groups

Course No: 
M304, M306
Review of Several variable Calculus: Directional Derivatives, Inverse Function Theorem, Implicit function Theorem, Level sets in R n , Taylor’s theorem, Smooth function with compact support. Manifolds: Differentiable manifold, Partition of Unity, Tangent vectors, Derivative, Lie groups, Immersions and submersions, Submanifolds. Vector Fields: Left invariant vector fields of Lie groups, Lie algebra of a Lie group, Computing the Lie algebra of various classical Lie groups. Flows: Flows of a vector field, Taylor’s formula, Complete vector fields. Exponential Map: Exponential map of a Lie group, One parameter subgroups, Frobenius theorem (without proof). Lie Groups and Lie Algebras: Properties of Exponential function, product formula, Cartan’s Theorem, Adjoint representation, Uniqueness of differential structure on Lie groups. Homogeneous Spaces: Various examples and Properties. Coverings: Covering spaces, Simply connected Lie groups, Universal
covering group of a connected Lie group. Finite dimensional representations of Lie groups and Lie algebras.
Reference Books: 
  1. D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
  2. S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, Graduate Studies in Mathematics 34, American Mathematical Society, 2001.
  3. S. Kumaresan, “A Course in Differential Geometry and Lie Groups”, Texts and Readings in Mathematics 22, Hindustan Book agency, 2002.
  4. F. W. Warner, “Foundations of Differentiable Manifolds and Lie Groups”, Graduate Texts in Mathematics 94, Springer-Verlag, 1983.

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