= Topological transition map
{title2=$g_{ji}=\varphi_j\circ\varphi_i^{-1}$}
For two <coordinate charts> in a <topological atlas>, the topological transition map is the <homeomorphism>
$$
g_{ji}:\varphi_i(U_i\cap U_j)\longrightarrow\varphi_j(U_i\cap U_j),\qquad g_{ji}=\varphi_j\circ\varphi_i^{-1}.
$$
Both domains are <open sets> in <Euclidean space>. <Function composition> gives $g_{ij}=g_{ji}^{-1}$ and $g_{ki}=g_{kj}\circ g_{ji}$ wherever all three <coordinate charts> overlap. These identities express consistency of the coordinate descriptions; imposing <smooth functions> as the coordinate changes gives a <smooth atlas>.
Back to article page
