Yu.Burman

Geometry of manifolds and bundles

• 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.
• 1-form.
• Pullback of a form.
• Differential of a function; expact 1-form.
• Closed 1-form.
• Integral of a 1-form over a curve.

Principal statements.

• The subset I_a Б 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 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.