### Geometries: Diffeomorphism Classes vs Quilts

#### Posted by John Baez

What follows is a guest post by Greg Weeks. If your memory extends back before the formation of this blog to the glory days of sci.physics.research, you should remember Greg.

*guest post by Greg Weeks*

Our topic: general relativity, manifolds vs geometries, and a conceivably new-and-improved definition of geometries

The new definition is so simple and obvious that I suspect that it is flawed or of no practical value. But, so far, I find it conceptually pleasing.

Our starting point is a Riemannian manifold $(M, g)$. (We exclude additional fields for simplicity.) $M$ is a point-set. The atlas making $M$ into a manifold is implicit. And you know what $g$ is.

The established definition of the **geometry** of $(M, g)$ is the set of all results of applying an active diffeomorphism, i.e.:

$\{ (M, \varphi_*(g)) : \varphi is \; \mathrm{a} \; diffeomorphism \; of M \}$

($\varphi_*$ is the universal “push forward” operator on tensor fields, induced by the diffeomorphism $\varphi$.)

The established definition is algebraic and lacks a point-set. The following definition is analytical and has a derived point-set.

Let $x:U \to \mathbb{R}^n$ be a chart on $(M, g)$. Then the Riemannian manifold

$(x(U), x_* (g|U))$

is isomorphic to $(U, g|U)$.

I call $(x(U), x_*(g|U))$ a “patch”. A patch describes a piece of the geometry of $(M, g)$. To obtain the entire geometry, we simply need to sew overlapping patches together.

*That’s all there is to it!* The rest is just details.

But before the details, I should stress the following:

Patches (and the sewing described below) live entirely on $\mathbb{R}^n$. The geometry makes no reference to $M$ or $g$. There is no a priori point-set and no diffeomorphism gauge group. On the other hand, there will be a *derived* point-set.

Now for the details.

First, we derive patches from *all* of the charts — for the same reason that manifolds are defined to have maximal atlases.

Unfortunately, patches derived from different charts can be identical, so we must tag them to keep them unique. (Yucch!) The tag values are irrelevant, so long as they are different for each chart. If we denote the tag associated with the chart $x$ by $tag(x)$, then the derived **patch** is the triple $(x(U), x_*(g|U), tag(x))$.

Second, we must sew overlapping patches together. For any pair of charts $x:U \to \mathbb{R}^n$ and $y:V \to \mathbb{R}^n$ and their derived patches

$p_x = (x(U), x_*(g|U), tag(x))$

and

$p_y = (y(V), y_*(g|V), tag(y)),$

we define the (possibly empty) overlap bijection

$B(p_x, p_y): x(U \cap V) ➝ y(U \cap V)$

by $B(p_x, p_y)(c) = y(x^{-1}(c))$.

We’re done. $B$ handles all of our sewing. The geometry is therefore $(P, B)$, where P is the set of all patches.

The geometry of $(M,\varphi_*(g))$ had better be the same as the above! And it is. The patches and overlap bijection derived from $x \circ \varphi^{-1}:φ(U)\to \mathbb{R}^n$ and $y \circ \varphi^{-1}:\varphi(V)\to \mathbb{R}^n$ on $(M, \varphi_*(g))$ are the same as those derived from $x:U\to\mathbb{R}^n$ and $y:V\to \mathbb{R}^n$ on $(M, g)$ — provided that you kindly choose the same tags.

Here’s an attempt to define geometries without any reference to manifolds.

A **geometry** is a pair $(P,B)$ that satisfies #1-5 below. Please keep in mind that #1-5 are requirements, not implications.

1. P is a set of **patches**. A patch is a triplet: an open subset of $\mathbb{R}^n$, a (smooth) Riemannian metric on the open set, and a unique tag. Given a patch $p$, we will use the notation $p.S$, $p.g$, and $p.tag$ for the three components.

2. $B$ is a function-valued function on $(P \times P)$. For every pair of patches $p_1$ and $p_2$, $B(p_1, p_2)$ is a (possibly empty) smooth bijection between open subsets of $p_1.S$ and $p_2.S$.

Suppose that $c_1 \in p_1.S$ and $c_2 \in p_2.S$. We refer to the pairs $(p_1, c_1)$ and $(p_2, c_2)$ as **patch points**. If $B(p_1, p_2)(c_1) = c_2$, then we say that the two patch points are **“the same”**. (This is a handy abuse of language. “the same” will always appear in quotes.)

Being “the same” is required to be an equivalence relation:

- $B(p, p)$ = the identity map on $p.S$
- $B(p_1, p_2) = B(p_2, p_1)^{-1}$
- If $B(p_1, p_2)(c_1) = c_2$ and $B(p_2, p_3)(c_2) = c_3$, then $B(p_1, p_3)(c_1) = c_3$.

3. Given “the same” patch points $(p_1, c_1)$ and $(p_2, c_2)$, then the components of $p_1.g(c_1)$ and $p_2.g(c_2)$ are related tensorially, using the partial derivatives of $B(p_1, p_2)$ and its inverse.

4. Given $p_1$ and $p_2$, if $p_1.S = p_2.S$ and $B(p_1, p_2)$ is the identity on $p_1.S$, then $p_1 = p_2$.

5. No further patches can be “added” — you know what I mean — whose patch points are all “the same” as pre-existing patch points. (This requirement is intended to achieve maximality.)

While the above definition of geometries does not have an a priori point-set, it does have a derived point-set. The points of the geometry are equivalence classes of patch points that are “the same”.

I suspect that we can create a manifold from a geometry by “pulling up” a topology, atlas, and Riemannian metric from the patches to the derived point-set.

Above, the tensor field $g$ could be replaced with any desired set of tensor fields. General relativity is implemented on a geometry by requiring that $G = \kappa T$ holds for every patch. I would hope that physically equivalent solutions are identical (modulo tags).

I would also hope that there is no Cauchy ambiguity. Take a complete solution and tear off and discard its later half. (Please make that well-defined somehow.) Then the equations of motion should uniquely grow back the discarded half (modulo tags), like a lizard growing back its tail.

So I hope.

## Re: Geometries: Diffeomorphism Classes vs Quilts

In particular it could be discarded altogether… in which case the discussion is about the definition of smooth manifolds as such. My understanding is that the proposal here is to observe that instead of saying

we could say

without mentioning the set of points. Because, as you observe, that set of points is directly recovered from the second statement as the set of equivalence classes of points in the patches.

I am not sure I see what the big difference between the two formulations is. (But maybe I misunderstand the proposal after all.)

I am also not sure yet that I understand what the proposal has to do with what John said in the introduction:

What is the “alternate approach” to talk about diffeomorphism classes of Riemannian manifolds? I don’t see this discussed in the above. Already the equivalence relation between different choices of atlases on the same manifold seems not to be discussed yet!?