Yu.Burman
Geometry of manifolds and bundles
In this course we introduce lots of notions. Among them are:

Manifold (with/without boundary)

Smooth mapping

Tangent space

Derivative (tangent mapping) of a smooth mapping

Bases d/dx_{i} and dx_{i}.

Vector bundle, oriented vector bundle.

Morphism of bundles.

Dual bundle.

Vector field.

Integral trajectory of a vector field.

Commutator of vector fields.

1form.

Pullback of a form.

Differential of a function; expact 1form.

Closed 1form.

Integral of a 1form over a curve.
Some of thes notions have several definitions; their equivalence is proved.
Principal statements.
Most statements proved in the course are about equivalence of various definitions.
Others are:

The subset I_{a} × Func(M) is a maximal ideal.

If M is compact then every maximal ideal in Func(M) is I_{a}.

Hadamard's lemma.

The fiber of a dual bundle is a space dual to the fiber of the original
bundle.

[X,fY]=f[X,Y]+X(f)Y.

Integral of a 1form over a curve is homotopy invariant if and only if
the form is closed.

X(<m,Y>)Y(<m,X>)=<m,[X,Y]> if and only if the form m is closed.