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Just as there are various types of manifolds, there are various types of maps of manifolds. PDIFF serves to relate DIFF and PL, and it is equivalent to PL.. In geometric topology, the basic types of maps correspond to various categories of manifolds: DIFF for smooth functions between differentiable manifolds, PL for piecewise linear functions between piecewise linear manifolds, and TOP for ...
Let : be a smooth map of smooth manifolds. Given , the differential of at is a linear map : from the tangent space of at to the tangent space of at (). The image of a tangent vector under is sometimes called the pushforward of by .
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the theory of harmonic maps. [1] [2] [3]
Pushforward (differential), the differential of a smooth map between manifolds, and the "pushforward" operations it defines; Pushforward (homology), the map induced in homology by a continuous map between topological spaces; Pushforward measure, measure induced on the target measure space by a measurable function
When the map between manifolds is a diffeomorphism, that is, it has a smooth inverse, then pullback can be defined for the vector fields as well as for 1-forms, and thus, by extension, for an arbitrary mixed tensor field on the manifold. The linear map = (, ())
Smooth maps between manifolds induce linear maps between tangent spaces: for :, at each point the pushforward (or differential) maps tangent vectors at to tangent vectors at (): ,: (), and on the level of the tangent bundle, the pushforward is a vector bundle homomorphism: :.
A map is a local diffeomorphism if and only if it is a smooth immersion (smooth local embedding) and an open map. The inverse function theorem implies that a smooth map f : X → Y {\displaystyle f:X\to Y} is a local diffeomorphism if and only if the derivative D f x : T x X → T f ( x ) Y {\displaystyle Df_{x}:T_{x}X\to T_{f(x)}Y} is a linear ...
For a particular smooth manifold or smooth map between smooth manifolds, this can be shown by breaking up the manifold into small enough pieces, and then linearizing the manifold or map on each piece: for example, a circle in the plane can be approximated by a triangle, but not by a 2-gon, since this latter cannot be linearly embedded.