Analysis on Manifolds

Analysis on Manifolds is an important extension of Multivariable Analysis. The main concepts and ideas in this theory are independent of the choice of coordinates. The course leads directly to de Rham cohomology and the Stokes formula.

Prerequisites: Multivariable Calculus or Multivariable Analysis, and Linear Algebra (vectors and linear forms, linear transformations and determinant).

Curriculum:

1. Definition and examples of smooth manifolds.
2. Tangent vectors and tangent space to a manifold at a point. Tangent bundles. Vector fields.
3. Skew-symmetric forms on linear spaces. Wedge product.
4. Differential forms on manifolds. Exterior differential.
5. Smooth maps of manifolds. Diffeomorphisms. The transformation rule under coordinate change for functions, vector fields and differential forms.
6. Orientability and orientation.
7. Integration in Rⁿ. Coordinate change in the integral. Integration of differential forms. Stokes theorem for a cube in Rⁿ.
8. Integration on manifolds.
9. Manifold with boundary. Induced orientation of the boundary.
10. General Stokes theorem. Green’s theorem, the divergence theorem, Stokes formula for a surface in Rⁿ.
11. Closed and exact differential forms. The Poincaré lemma. De Rham cohomology.

Textbooks

• J.Munkres, Analysis on Manifolds.
• M.Spivak, A Comprehensive Introduction to Differential Geometry. Volume One.
• J.Lee, Introduction to Smooth Manifolds.