Chart Transition Maps

Where the open sets of two charts \((U,x)\) and \((V,y)\) overlap, each chart assigns coordinates to every point of the overlap. The chart transition map is the conversion between them: it takes the tuple \(x(p)\) to the tuple \(y(p)\). Its domain and its target are open subsets of \(\mathbb{R}^d\), so the space itself is no longer in it.

An element of \(x(U \cap V)\) is \(x(p)\) for exactly one \(p \in U \cap V\), since \(x\) is invertible on \(U\). Applying \(x^{-1}\) recovers \(p\), and applying \(y\) gives that chart's tuple for it. The composite takes the tuple assigned by the first chart to the tuple assigned by the second.

Each half of the composition restricts it. \(x^{-1}\) is defined on \(x(U)\), so no more than that can be fed in. \(y\) is defined on \(V\), so whatever \(x^{-1}\) returns must lie in \(V\). Together these force the input to be \(x(p)\) for a \(p\) lying in both \(U\) and \(V\), which is exactly the set \(x(U \cap V)\).

The set \(x(U \cap V)\) is open in \(\mathbb{R}^d\), because \(U \cap V\) is open in the space and \(x\) carries open sets of \(U\) to open sets of \(x(U)\), which are open in \(\mathbb{R}^d\) since \(x(U)\) is.

The space appears in the construction of the map and nowhere in its statement: a transition map assigns \(d\)-tuples to \(d\)-tuples.

What it returns is chart-dependent: replace either chart and both the tuples and the assignment between them can change.

Reversing the roles of the two charts gives \(x \circ y^{-1} : y(U \cap V) \to x(U \cap V)\), and the two compose to the identity in both orders: following \(y \circ x^{-1}\) by \(x \circ y^{-1}\) returns \(x(p)\) to itself, and likewise the other way around.

Both are continuous, since each is a composition of continuous maps: \(x^{-1}\) and \(y^{-1}\) are continuous because a chart map is a homeomorphism, and \(x\) and \(y\) are continuous for the same reason. A transition map is therefore a homeomorphism between two open subsets of \(\mathbb{R}^d\).

Let \(M\) be the unit circle, \((U,x)\) the angle chart taking values in \((0,2\pi)\) that omits \((1,0)\), and \((V,y)\) the angle chart taking values in \((-\pi,\pi)\) that omits \((-1,0)\). The overlap \(U \cap V\) is the circle without both points: an upper arc and a lower arc. Its image is \(x(U \cap V) = (0,\pi) \cup (\pi, 2\pi)\).

For \(a \in (0,\pi)\), the point \(x^{-1}(a)\) sits on the upper arc and \(y\) returns the same angle, so \((y \circ x^{-1})(a) = a\). For \(a \in (\pi, 2\pi)\), the point sits on the lower arc and \(y\) measures it negatively: \((y \circ x^{-1})(a) = a - 2\pi\). Those two formulas are the whole map. At \(q = (0,-1)\): \(x(q) = 3\pi/2\), and \((y \circ x^{-1})(3\pi/2) = 3\pi/2 - 2\pi = -\pi/2\), which is \(y(q)\).