
Textbooks, References, and Topics
Course Text
 R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensors, Analysis, and Applications, Second Edition. SpringerVerlag, 1988. (Draft Third Edition, Current Update started January 3, 2007)
Note: Links go to version cds20208
Lecture Notes and Supplements
Additional references that may be useful:
 Anthony Bloch, Nonholonomic Mechanics and Control, Springer Verlag.
 W. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. Academic Press, 1986.
 V. Guillemin and A. Pollak, Differential Topology, PrenticeHall, 1974.
 J.M. Lee, Introduction to Smooth Manifolds, SpringerVerlag, 2002.
 J. Milnor, Topology From the Differentiable Viewpoint, University Press of Virginia, 1965.
 B. Schutz, Geometrical Methods of Mathematical Physics, Cambridge University Press, 1980.
 M. Spivak, A Comprehensive Introduction to Differentiable Geometry, vol I. Publish or Perish, 1970.
 F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, SpringerVerlag, 1983.
Course Topics
 Motivation for geometric methods: general, historic introduction; example of pendulum: introduce manifolds as configuration spaces; differentiation leads to notion of tangent space; vector fields on manifolds as fundamental object; and, Lie groups as symmetry groups.
 Topology: motivation; metric spaces; topological spaces; mappings between topological spaces; properties.
 Differential manifolds: topological manifolds; charts; manifolds; examples; constructing manifolds; open submanifold; product manifolds.
 Mappings between manifolds: diffeomorphisms, submersions, immersions.
 Submanifolds: open, immersed, imbedded, and regular submanifolds; mention Whitney Imbedding Theorem.
 Tangent space: link geometric intuition with formal definition; pullback; pushforward; coordinate transformations.
 Tangent bundle: fiber bundles, sections; distributions.
 Vector fields: integral curves; flow of a vector field; coordinatefree definition of dynamical system.
 Lie algebras: Lie derivative; JacobiLie bracket; Lie algebras; role in control of nonlinear systems.
 Frobenius theorem
 Lie groups
 Covector fields: dual of vector space; covectors and covector fields; pullback of covector field.
 Lie derivatives
 Tensors
 Riemannian manifolds
 Exterior algebra
 Manifolds with boundaries
 Integration on manifolds
 Discrete exterior calculus
