B1 Course. Smooth Manifolds and Observables

General Introduction

• Observation mechanism in classical physics.
• States of a classical system.
• Commutative algebras and their homomorphisms.
• Spectra of commutative algebras.
• Geometric, complete, and smooth algebras.
• Ideals of points and submanifolds.
• Localization of algebras to open subsets of the spectrum.
• Spectral theorem: equivalence of the algebraic and geometric definitions. of smooth manifolds
• Smooth maps of manifolds and homomorphisms of smooth algebras.

First Order Differential Calculus on Manifolds Algebraically

• Tangent vectors and vector fields.
• Relative vector fields.
• The functor D.
• Behaviour of tangent vectors and vector fields with respect to smooth mappings of manifolds.
• The flow of a vector field.
• Lie derivative of vector fields.
• The Lie algebra of vector fields.
• Covectors and differential forms.
• Tensors and their operations.
• The algebra of differential forms.
• Behaviour of differential forms and covariant tensors with respect to differentiable mappings of manifolds.
• Lie derivatives of covariant tensors.
• Exterior differential and the de Rham cohomology.
• Cartan’s formula and homotopy property of de Rham cohomology.
• Distributions and the Frobenius theorem.
• Cohomological theory of integration.

References

• Jet Nestruev, Smooth manifolds and Observables, Springer-Verlag, Graduate Texts in Mathematics, Vol. 220 (2002).