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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]
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.
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: :.
Given any two local parametrizations f : V → U and f ′ : V ′→ U ′ of a regular surface, the composition f −1 ∘ f ′ is necessarily smooth as a map between open subsets of ℝ 2. [17] This shows that any regular surface naturally has the structure of a smooth manifold, with a smooth atlas being given by the inverses of local ...
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.
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 ...
In mathematics, particularly topology, an atlas is a concept used to describe a manifold. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fiber bundles.