Yu.Burman
Geometry of manifolds and bundles
In this course we introduce lots of notions. Among them are:
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Manifold (with/without boundary)
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Smooth mapping
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Tangent space
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Derivative (tangent mapping) of a smooth mapping
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Bases d/dxi and dxi.
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Vector bundle, oriented vector bundle.
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Morphism of bundles.
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Dual bundle.
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Vector field.
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Integral trajectory of a vector field.
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Commutator of vector fields.
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1-form.
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Pullback of a form.
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Differential of a function; expact 1-form.
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Closed 1-form.
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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:
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The subset Ia × Func(M) is a maximal ideal.
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If M is compact then every maximal ideal in Func(M) is Ia.
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Hadamard's lemma.
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The fiber of a dual bundle is a space dual to the fiber of the original
bundle.
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[X,fY]=f[X,Y]+X(f)Y.
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Integral of a 1-form over a curve is homotopy invariant if and only if
the form is closed.
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X(<m,Y>)-Y(<m,X>)=<m,[X,Y]> if and only if the form m is closed.