Differential Geometry Basic notions of manifold theory. Exterior …

Short Description
Main theorem of ODE (the theory of ordinary differential equations). Frobenius theorem. Exterior differential calculus and integration theory. …

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Content
Differential Geometry
Joseph Bernstein Fall 2006
This is a basic course on differential geometry for toar rishon.
Preliminary material.
1. Reminder of basic notions of linear algebra over R.
Linear space, basis, dimension, dual space.
Quadratic forms. Euclidean spaces.
2. Metric and topology on the Euclidean space Rn.
Basic notions of metric spaces and topological spaces.
Local and global properties.
3. Differentiable functions on Rn. Differentiable morphisms. Differentials.
4. Curves in Euclidean space Rn. Curvature and torsion of a curve.
Frenet formulas.
Basic notions of manifold theory.
5. Notion of a smooth manifold. Morphisms of manifolds.
Cut-off functions and partition of unity.
Local to global correspondence.
6. Inverse and implicit function theorems.
Immersions, submersions, morphisms of constant rank.
Submanifolds
7. Notion of tangent and cotangent spaces. Differentials.
8. Vector fields. Commutator.
Main theorem of ODE (the theory of ordinary differential equations).
Frobenius theorem.
Exterior differential calculus and integration theory.
9. Differential forms. De Rham differential. Functoriality.
Lie derivative, Weyl formulas.
10. Integration of 1-forms.
Orientation. Integration of differential forms. Change of variables.
Manifolds with boundary. Stokes’ formula.
Classical formulations.
Applications of Stokes’ formula.
11.General remarks on cohomologies. DeRham cohomology.
Poincare lemma and DeRham theorem.
Classical theory…