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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.
A candlestick chart of the Euro against the USD, marked up by a price action trader. A price action trader's analysis may start with classical price action technical analysis, e.g. Edwards and Magee patterns including trend lines, break-outs and pullbacks, [13] which are broken down further and supplemented with extra bar-by-bar analysis, sometimes including volume.
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the -sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifolds.
Pushforward (differential) In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near .
Vector bundle. The (infinitely extended) Möbius strip is a line bundle over the 1-sphere S1. Locally around every point in S1, it looks like U × R (where U is an open arc including the point), but the total bundle is different from S1 × R (which is a cylinder instead). In mathematics, a vector bundle is a topological construction that makes ...