Textbooks, References, and Topics

Course Text

Note: Links go to version cds202-08

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, Prentice-Hall, 1974.

  • J.M. Lee, Introduction to Smooth Manifolds, Springer-Verlag, 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, Springer-Verlag, 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; pull-back; push-forward; coordinate transformations.

  • Tangent bundle: fiber bundles, sections; distributions.

  • Vector fields: integral curves; flow of a vector field; coordinate-free definition of dynamical system.

  • Lie algebras: Lie derivative; Jacobi-Lie bracket; Lie algebras; role in control of nonlinear systems.

  • Frobenius theorem

  • Lie groups

  • Covector fields: dual of vector space; covectors and covector fields; pull-back of covector field.

  • Lie derivatives

  • Tensors

  • Riemannian manifolds

  • Exterior algebra

  • Manifolds with boundaries

  • Integration on manifolds

  • Discrete exterior calculus