Transition Maps
A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)
To be more precise, suppose that and are two charts for a manifold M such that is non-empty. The transition map is the map defined by
Note that since and are both homeomorphisms, the transition map is also a homeomorphism.
Read more about this topic: Atlas (topology)
Famous quotes containing the words transition and/or maps:
“Some of the taverns on this road, which were particularly dirty, were plainly in a transition state from the camp to the house.”
—Henry David Thoreau (1817–1862)
“And now good morrow to our waking souls,
Which watch not one another out of fear;
For love all love of other sights controls,
And makes one little room an everywhere.
Let sea-discoverers to new worlds have gone,
Let maps to other, worlds on worlds have shown,
Let us possess one world; each hath one, and is one.”
—John Donne (1572–1631)