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When =, then the pullback and the pushforward describe the transformation properties of a tensor on the manifold . In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.
Then, yields, in the above sense, the pushforward φ ∗ X, which is a vector field along φ, i.e., a section of φ ∗ TN over M. Any vector field Y on N defines a pullback section φ ∗ Y of φ ∗ TN with (φ ∗ Y) x = Y φ(x).
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. Pushforward (differential) , the differential of a smooth map between manifolds, and the "pushforward" operations it defines
In general, any measurable function can be pushed forward. The push-forward then becomes a linear operator, known as the transfer operator or Frobenius–Perron operator.In finite spaces this operator typically satisfies the requirements of the Frobenius–Perron theorem, and the maximal eigenvalue of the operator corresponds to the invariant measure.
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 ...
Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X × B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
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.
Conversely, if f is proper, for Y a subvariety of X the pushforward is defined to be ([]) = [()] where n is the degree of the extension of function fields [k(Y) : k(f(Y))] if the restriction of f to Y is finite and 0 otherwise. By linearity, these definitions extend to homomorphisms of abelian groups