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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 ...
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]
If a map, φ, carries every point on manifold M to manifold N then the pushforward of φ carries vectors in the tangent space at every point in M to a tangent space at every point in N. In differential geometry , pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.
This category includes maps between manifolds, smooth or otherwise, particularly in geometric topology. Pages in category "Maps of manifolds" The following 14 pages are in this category, out of 14 total.
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 = (, ())
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
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 ...
Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives. If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth.