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MA455 Content

Content: Smooth manifolds are generalizations of the notion of curves and surfaces in  \mathbb {R}^{3} and provide a rigorous mathematical concept of space as well as a natural setting for analysis. They form a fundamental part of modern mathematics and are used widely in pure and applied subjects such as differential geometry, general relativity and partial differential equations.

We begin the module with the definition of an abstract smooth manifold and vector bundles, in particular, the tangent and cotangent bundle. We will introduce submanifolds and learn to use the implicit function theorem to construct such objects and learn how manifolds can be concretely realized as subsets of Euclidean space. We will also study Lie groups and their algebras as examples of manifolds and how to construct manifolds using group actions. Next, we return to the tangent bundle to study vector fields and their integral curves. We continue with differential forms and Stokes theorem, the generalization to manifolds of the divergence theorem and touch on de Rham cohomology. Then we consider integrable distributions and the Frobenius theorem.

Aims: To introduce notion of an abstract smooth manifold and to develop students geometric intuition of manifolds together with rigorous analysis.

Objectives: To introduce students to some important ideas that underlie the theory of differentiable manifolds and to develop skills in manipulation of differentiable objects;

  1. Construction of manifolds using the implicit function theorem and quotients by group actions;
  2. Manipulation of differential forms and Stokes theorem;
  3. Vectors fields as local infinitesimal generators and the integration of flows;
  4. Frobenius theorem.

Books:

The course text is:

Lee, J .M. Introduction to Smooth Manifolds, Springer-Verlag;

Additional texts:

Warner, F. Foundations of differentiable manifolds and Lie groups, Springer-Verlag;

Boothby, W. An introduction to differentiable manifolds and Riemannian geometry, Academic Press;