Course Text
- R. Abraham, J. E. Marsden, and T. Ratiu,
Manifolds, Tensors, Analysis, and Applications
, Second Edition. Springer-Verlag, 1988.
(Draft Third Edition, Last Update February 10, 2004)
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. 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. Lie groups as symmetry groups.
- Topology: motivation; metric spaces; topological spaces; mappings
between topological spaces; properties.
- Differential manifolds: topological manifolds; charts;
$C^infty$-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
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