Let and be two coordinate patches on an -manifold . We can restrict each coordinate map to the intersection of the two patches — if the intersection is empty this is trivial, but not wrong — and get a function from one open region of to another. Specifically, if the coordinate patches have images and , then we get a function . We call this the “transition function” from one coordinate patch to another, and often write it
So what do we know about these transition functions? Well, is a homeomorphism from to , meaning it’s continuous with a continuous inverse. But remember that , and we know a lot about real -dimensional space. We know all kinds of extra structure beyond just its topology, and in particular we know what it means for a function from to itself to be smooth.
Now, as I’ve said before, “smooth” ends up behaving like a term of art. Really what we know is what it means for a function from to itself to be differentiable, and for that differential to be continuous, and for there to be a second differential, and so on. We even introduced classes of functions to describe this whole tower, where consists of continuous functions, consists of continuously differentiable functions, consists of continuously twice-differentiable functions. This pattern continuous on through for continuously -times-differentiable functions, for functions with arbitrarily many continuous derivatives, and for “analytic” functions which are the limits of their own Taylor series.
One case we didn’t explicitly mention is : the class of “piecewise linear” functions. These are continuous functions that can be defined by breaking up their domains into a finite number of chunks and giving an affine function — a linear transformation plus a constant offset vector — on each chunk so that the borders line up continuously. These functions aren’t generally differentiable, since they turn sharp corners at the boundaries of chunks, but they’re pretty nicely behaved anyway. The class sits between and .
Anyway, for each class of functions we can define a corresponding class of manifolds. If all the transition functions between all the coordinate patches are piecewise-linear, we say that we have a “PL-manifold”. If they’re all , we say that we have a -manifold or a “differential manifold”. This goes up through -manifolds, -manifolds, and analytic manifolds.
In practice, “smooth” ends up meaning , so that we can always take as many derivatives as we want, but really we rarely need more than a few. Still, we’ll say “smooth” and not worry about it. And from here on all our manifolds will be smooth unless explicitly stated otherwise.