Textbooks, References, and Topics

Course Text

• R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensors, Analysis, and Applications, Second Edition. Springer-Verlag, 1988. (Draft Third Edition, Current Update started January 3, 2007)

Lecture Notes and Supplements

• Differential Forms (pdf)
• Manifolds (pdf)

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