Manifolds and Lie Groups

**Instructor:**Guanghua JI

**Classroom and Time:**C501, 5-7

**Prerequisites:**General topology, functional analysis, group theory.

**Grading:**The course grade will be based homework(50%), final exam(50%).

**References:**

1, A. A. Sagle, R.E. Walde, Introduction to Lie groups and Lie Algebra, 1973.

2, Y. Matsushima, Differentiable Manifods, 1972.

3, F. Warner, Foundations of Differentiable Manifolds and Lie Groups, GTM 94.

4, Loring W. Tu, An Introduction to Manifolds, 2nd edition, Springer.

5, N. R. Wallach, Harmonic Analysis on Homogeneous Spaces, 1973.

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Syllabus:

**Chapter I, Differentiable Manifolds**

**3/11**,1.1, Differentiable manifolds:I, Definitions. II, Partitions of unity. III, Tangent vectors. IV. The differential map. V, Tangent and cotangen bundles. HW: 1, 3.

**3/18,**1.2, Submanifolds: I, Immersions and submersions. II, Inverse function thm. III, Submanifolds. IV, Implicit function thm. HW: 9,10.

**3/25,**1.3, Vector Fields:I, Definitions. II, The Lie bracket. HW: 12.

**4/01**, III, Intergral curves and local flows.1.4, The Frobenius theorem:I, Distributions. HW: 15,17,18.

**4/08,**II, Frobenius theorem. HW: 21,22.

Chapter II, Differential forms and Tensor fields,

**4/15**,2.1, Tensor fields and differential forms:I, Multilinear algebra: tansor algebra, exterior algebra, symmetric algebra. II,Tensor fields. III, Differential forms. HW: 2,5.

4/22,2.2, The Lie derivative:HW

**Chapter III, Lie Groups and Homogeneous spaces**

4/29,3.1, Lie Groups and their Lie algebra:I, Topology groups. II, Lie Groups. III, Lie Algebra of Lie groups. HW

5/6,3.2,The exponrntial map: I, One-parameter subgroups. II, The exponential map.

5/13,3.3, Homomorphisms and Lie subgroups

5/20,3.4, The Adjoint representation

5/27,3.5, Homogeneous manifolds

**Chapter IV, Integration on Manifods**

6/3,4.1, Integration of differential forms

6/10,4.2, Invariant integration of Lie groups

6/17,4.3, Stokes' Theorem

Chapter V, Representations Theory

**6/24,5.1, Elementary representations theory**

Manifolds and Lie Groups Final Exam