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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/dxi and dxi.
Vector bundle, oriented vector bundle.
Morphism of bundles.
Dual bundle.
Vector field.
Integral trajectory of a vector field.
Commutator of vector fields.
1-form.
Pullback of a form.
Differential of a function; expact 1-form.
Closed 1-form.
Integral of a 1-form 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 Ia в Func(M) is a maximal ideal.
If M is compact then every maximal ideal in Func(M) is Ia.
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 1-form 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.