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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 resulting new pullback bundle looks locally like a Cartesian product of the new base space, and the (unchanged) fiber.
More generally, any covariant tensor field – in particular any differential form – on may be pulled back to using . When the map ϕ {\displaystyle \phi } is a diffeomorphism , then the pullback, together with the pushforward , can be used to transform any tensor field from N {\displaystyle N} to M {\displaystyle M} or vice versa.
In this case E 1 × E 2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E 1 × B E 2, which is a fiber bundle over B. The pullback of two smooth transverse maps into the same differentiable manifold is also a differentiable manifold, and the tangent space of the ...
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In algebraic topology, given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism:
Ukraine on Sunday called on Belarus to pull back what it described as significant levels of Belarusian forces and equipment deployed at their common border. The Ukrainian foreign ministry warned ...
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain.
A pullback motor (also pull back, pull back and go or pull-back) is a simple clockwork motor used in toy cars. A patent for them was granted to Bertrand 'Fred' Francis in 1952 as a keyless clockwork motor. [1] [2] Pulling the car backward (hence the name) winds up an internal spiral spring; a flat spiral rather than a helical coil spring. When ...