+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Manifolds and cobordisms +-- {: .hide} [[!include manifolds and cobordisms - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/ChartsOfAManifold.png" width="400"> </div> A [[topological manifold]] $X$ of [[dimension]] $n \in \mathbb{N}$ is by definition a [[topological space]] that is locally [[homeomorphic]] to a [[Cartesian space]] $\mathbb{R}^n$. A choice of such morphism $$ \phi : \mathbb{R}^n \stackrel{\simeq}{\to} X|_{im(\phi)} \hookrightarrow X $$ is a **coordinate system** or **coordinate chart** or just **chart** on the [[image]] of $\phi$. This generalises to other sorts of [[manifolds]]. An [[atlas]] is the collection of coordinate charts defining a manifold structure. > graphics grabbed from [[The Geometry of Physics - An Introduction|Frankel]] ## Related concepts * [[coordinate transformation]], [[diffeomorphism]] * [[Riemann normal coordinates]], [[Walker coordinates]] * [[polar coordinates]] [[!redirects coordinate system]] [[!redirects coordinate systems]] [[!redirects chart]] [[!redirects coordinate chart]] [[!redirects charts]] [[!redirects coordinate charts]] [[!redirects local chart]] [[!redirects local coordinate chart]] [[!redirects local charts]] [[!redirects local coordinate charts]] [[!redirects coordinate]] [[!redirects coordinates]] [[!redirects coordinate patch]] [[!redirects coordinate patches]] [[!redirects coordinate function]] [[!redirects coordinate functions]] [[!redirects local coordinate]] [[!redirects local coordinates]]