Course No
M563
Credit
4
Approval
Syllabus
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
- D. Bump, “Lie Groups”, Graduate Texts in Mathematics 225, Springer, 2013.
- S. Helgason, “Differential Geometry, Lie Groups and Symmetric Spaces”, Graduate Studies in Mathematics 34, American Mathematical Society, 2001.
- S. Kumaresan, “A Course in Differential Geometry and Lie Groups”, Texts and Readings in Mathematics 22, Hindustan Book agency, 2002.
- F. W. Warner, “Foundations of Differentiable Manifolds and Lie Groups”, Graduate Texts in Mathematics 94, Springer-Verlag, 1983.