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Pullback (differential geometry) Let be a smooth map between smooth manifolds and . Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . This linear map is known as the pullback (by ), and is frequently denoted by .
Pullback (category theory) In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written.
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the ...
It leads to the existence of pullback maps in other situations, such as pullback homomorphisms in de Rham cohomology. Formally, let f : M → N be smooth, and let ω be a smooth k-form on N. Then there is a differential form f ∗ ω on M, called the pullback of ω, which captures the behavior of ω as seen relative to f.
In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold of dimension , a volume form is an -form. It is an element of the space of sections of the line bundle , denoted as . A manifold admits a nowhere-vanishing volume form if and only if it is ...
Inverse image functor. In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inverse image functor is a functor from the category of sheaves on Y to the category of sheaves on X.
The tautological bundle is also called the universal bundle since any vector bundle (over a compact space [1]) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes.
Formally, the normal bundle[2] to N in M is a quotient bundle of the tangent bundle on M: one has the short exact sequence of vector bundles on N: where is the restriction of the tangent bundle on M to N (properly, the pullback of the tangent bundle on M to a vector bundle on N via the map ). The fiber of the normal bundle in is referred to as ...