Maps of manifolds information

the category of piecewise-smooth maps between piecewise-smooth manifolds. In addition to these general categories of maps, there are maps with special...

(e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable...

classification of manifolds is a basic question, about which much is known, and many open questions remain. Low-dimensional manifolds are classified by...

The manifold absolute pressure sensor (MAP sensor) is one of the sensors used in an internal combustion engine's electronic control system. Engines that...

and Fréchet manifolds, in particular manifolds of mappings are infinite dimensional differentiable manifolds. For a Ck manifold M, the set of real-valued...

In the mathematical field of differential geometry, a smooth map between Riemannian manifolds is called harmonic if its coordinate representatives satisfy...

complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example...

( N , h ) {\displaystyle (N,h)} be Riemannian manifolds or more generally pseudo-Riemannian manifolds. An isometric embedding is a smooth embedding f...

equivalence between X and Y is a pair of continuous maps f : X → Y and g : Y → X, such that g ∘ f is homotopic to the identity map idX and f ∘ g is homotopic to...

class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are...

Lipschitz maps between such manifolds, similarly to how one defines smooth maps between smooth manifolds: if M and N are Lipschitz manifolds, then a function...

mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. Geometric topology as an...

analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that...

a Riemannian manifold, is a harmonic map if and only if it is a geodesic. If M and N are two Riemannian manifolds, then a harmonic map u : M → N {\displaystyle...

known as manifold learning, is any of various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with...

However, in statements involving maps between manifolds, one often has to restrict consideration to Fredholm maps, that is, maps whose differential at every...

partial differential equations, including the theory of harmonic maps. Given Riemannian manifolds M {\displaystyle M} and N {\displaystyle N} , which is...

study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical...

diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the...

distortion is an example of an ambient isotopy. More precisely, let N {\displaystyle N} and M {\displaystyle M} be manifolds and g {\displaystyle g} and...

Riemannian metrics) are all complete manifolds. All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete. All symmetric...

of the inverse. There are also versions of the inverse function theorem for complex holomorphic functions, for differentiable maps between manifolds,...

called such maps rotations) where note that A is not assumed to be a linear isometry. Then A maps midpoints to midpoints and is linear as a map over the...

function theorem to maps between manifolds. Coordinate-induced basis Cotangent space Differential geometry of curves Exponential map Vector space do Carmo...