These lecture notes were created using material from Prof. Helgason’s books *Differential Geometry, Lie Groups, and Symmetric Spaces* and *Groups and Geometric Analysis*, intermixed with new content created for the class. The notes are self-contained except for some details about topological groups for which we refer to Chevalley’s Theory of Lie Groups I and Pontryagin’s Topological Groups. Documenting the material from the course, the text has a fairly large bibliography up to 1978. Since then, a huge number of books on Lie groups has appeared.

### Source Textbooks

All excerpts courtesy of the American Mathematical Society. Used with permission.

Helgason, Sigurdur. *Differential Geometry, Lie Groups, and Symmetric Spaces*. Providence, R.I.: American Mathematical Society, 2001. ISBN: 0821828487.

Helgason, Sigurdur. *Groups and Geometric Analysis*. Providence, R.I.: American Mathematical Society, 2000. ISBN: 0821826735.

### Referenced Textbooks

Chevalley, Claude. *Theory of Lie groups, I*. Princeton, Princeton University Press, 1946.

Pontryagin, L. S. *Topological Groups*. Translated from the Russian by Arlen Brown, with additional material translated by P. S. V. Naidu. 3rd ed. New York: Gordon and Breach Science Publishers, 1986. ISBN: 2881241336 (Switzerland).

**Preface (**PDF**)**

**Chapter I: Elementary Differential Geometry (**PDF**)**

1. Manifolds

2. Mappings

3. Affine Connections

4. Parallelism

5. The Exponential Mapping

6. Covariant Differentiation

**Chapter II: Lie Groups and Lie Algebras (**PDF 1 of 2 - 1.9 MB**) (**PDF 2 of 2 - 1.8 MB**)**

1. The Exponential Mapping

2. Lie Subgroups and Subalgebras

3. Lie Transformation Groups

4. Coset Spaces and Homogeneous Spaces

5. The Adjoint Group

6. Semisimple Lie Groups

7. The Universal Covering Group

8. General Lie Groups

9. Differential Forms

10. Integration of Forms

11. Invariant Differential Forms

12. Invariant Measures on Coset Spaces

13. Real Forms of Complex Lie Algebras

14. The Classical Groups and their Cartan Involutions

**Chapter I: Exercises and Further Results (**PDF**)**

A. Manifolds

B. The Lie Derivative and the Interior Product

C. Affine Connections

D. Submanifolds

E. The Hyperbolic Plane

**Chapter II: Exercises and Further Results (**PDF**)**

A. On the Geometry of Lie Groups

B. The Exponential Mapping

C. Subgroups and Transformation Groups

D. Closed Subgroups

E. Invariant Differential Forms

F. Invariant Measures

G. Compact Real Forms and Complete Reducibility

**Solutions to Exercises (**PDF**)**