Difference between revisions of "Transition map"

Revision as of 06:20, 7 April 2015 (view source) Alec (Talk | contribs) (Created page with " ==Definition== Given two charts {{M|(U,\varphi)}} and {{M|(V,\psi)}} on a topological {{M|n-}}manifold where {{M|U\cap V\ne\emptyset}}<ref>Introduction to smooth ma...")		Latest revision as of 06:33, 7 April 2015 (view source) Alec (Talk | contribs) m
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	==Extending to smooth structures==		==Extending to smooth structures==
−	See [[Smoothly compatible]]	+	See [[Smoothly compatible charts]]
	==See also==		==See also==
	* [[Chart]]		* [[Chart]]
−	* [[Smoothly compatible]]	+	* [[Smoothly compatible charts]]
	* [[Topological manifold]]		* [[Topological manifold]]

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Latest revision as of 06:33, 7 April 2015

Definition

Given two charts $(U,\varphi)$ and $(V,\psi)$ on a topological $n-$manifold where $U\cap V\ne\emptyset$^[1] a transition map allows us to move from local coordinates of $\varphi$ to local coordinates of $\psi$ as the picture on the right shows.

The transition map, $\tau$ is defined as follows:

$$\tau:\varphi(U\cap V)\rightarrow\psi(U\cap V)$$ given by $$\tau=\psi\circ\varphi^{-1}$$

$\tau$ is a Homeomorphism because both $\varphi$ and $\psi$ are homeomorphisms, making $\tau$ a chart, $(U\cap V,\tau)$

Extending to smooth structures

See Smoothly compatible charts

See also

• Chart
• Smoothly compatible charts
• Topological manifold

References

1. Jump up ↑ Introduction to smooth manifolds - John M Lee - Second Edition