Better Glue for Manifolds

After my [initial post about manifolds](http://scienceblogs.com/goodmath/2006/10/manifolds_and_glue.php), I wanted to say a bit more about gluing.

You can form manifolds by gluing manifolds with an arbitrarily small overlap - as little as a single point along the point of contact between the manifolds. The example that I showed, forming a spherical shell out of two circles, used a minimal overlap. If all you want to do is show that the topology you form is a manifold, that kind of trivial gluing is sufficient, and it's often the easiest way to splice things together.

But there are a lot of applications of manifolds where you need more than that. So today, I'm going to show you how to do proper gluing in a way that preserves things like metric properties of manifolds when they're glued together.

Why isn't the trivial single-point overlap a sufficient basis for gluing manifolds?

There are lots of applications where you need more information about what happens when you cross the glue border. For example, to be able to do calculus on the surface of a manifold, you need to be able to define a metric for the entire surface - and the single-point overlap isn't enough to let you unambiguously define what the metric for the new manifold should be. As a simple example, suppose we were doing the simple gluing of two circles. In their metrics, one of the circles has radius three, and one has radius five. How long is an arc segment between the centers of the two circles? The answers to questions like that become quite awkward when you have a trivial overlap in your patchwork.

So most of the time, when you form manifolds by gluing, you create very non-trivial overlaps. For example, the classic way of forming a circle is using *four* open line segments - each one is which is mapped to a full half of the circle:

In this, only the precise center of each of the segments is *not* part of any overlap. There are four invertible continuous functions to define the gluing. The figure illustrates one of them - for the left-hand and top segments. The gluing defines how each of the segments maps onto a half of the circle; and the overlap is defined by functions mapping matching halves of the segments. In the example, the purple segment *p* overlaps with the green segment *g* in the upper left quadrant. The function that defines the overlap, f : *p₁* → *g₁*, maps between the upper half of the purple segment and the left half of the green segment.

This large overlap allows us to easily define a meaningful metric for the circle. The mapping in the overlap range provides a *coordinate transform* - a precise and unambiguous way to defining how a distance on *p* can be translated to a distance on *g*. we can pick either metric - the *p* metric or the *g* metric, and use it over all of both *p* and *g*. Just take a line segment that fits entirely into the overlap; measure it in *p*, and measure it in *g*, and you've got a ratio that does the job. (It's actually a bit more complicated than that, but that should give you the sense of it.)

And once we know the transformation from the *g* metric to the *p* metric, we can use the coordinate transformation for the overlap between *g* and the red segment to define a transformation from *p* to red's metric. So we can assign a single, unambiguous uniform metric to the entire circle.