On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces
Posted by Simon Willerton
Back in my recent post on intrinsic volumes I said that I was just finishing a paper on the magnitude of certain Riemannian manifolds. It is now essentially finished.
I will put it on the arxiv next week: comments and suggestions are welcome.
There are two main results in the paper. The first is a calculation of the magnitude of the $n$-sphere and the second is the asymptotic behaviour of the magnitude for an arbitrary homogeneous Riemannian manifold in terms of volume and total scalar curvature, a special case of the latter is the asymptotic behaviour of the magnitude of a homogeneous surface. I will explain these below, but first remind you a little about magnitude.
On the Magnitude…
Magnitude of finite metric spaces (under the name cardinality) was introduced by Tom Leinster at this very café, by analogy with his notion of Euler characteristic of a category. In two posts, Entropy, Diversity and Cardinality (Part 1) and Part 2, he explained how this was related to ideas in ecology and went on to prove a result on how magnitude helps you maximize biological diversity. In another direction Tom and I defined the magnitude of various infinite subsets of Euclidean space by approximating these with finite metric spaces. In the case of a finite metric space, as the space is scaled up, the magnitude gets closer and closer to the number of points: in the case of some nice subsets of Euclidean space we observed that as the subset is scaled up, the magnitude seems to tend to some linear combination of classical ‘intrinsic volumes’ of the subset.
[If you want some idea of the intuition I use to think of the magnitude then you should look at the post More Magnitude of Metric Spaces and Problems with Penguins.]
In this paper I take a different tack in defining the magnitude of infinite metric spaces, one that was suggested to me by Tom Leinster and by David Speyer. This involves using a (signed) measure on a metric space, and it clearly generalizes the idea of having a weight associated to each point in a finite metric space. Using this approach gives the same answer as we got in examples we’d previously calculated.
A key observation, essentially made by David Speyer here, is that for a homogeneous space $X$ (i.e., one that looks the same at every point) the magnitude can be calculated using any measure $\mu$ by the formula
$\left|X\right|= \frac{\int_{x\in X} d\mu}{\int_{x\in X}e^{-d(x,y)} d\mu},\qquad\text{for any} y\in X$
provided that the denominator is finite and non-zero. Aside from working with subsets of the line, this is currently the only way I know of calculating the magnitude of infinite metric spaces.
…of Spheres,…
There are two obvious metrics to consider on the $n$-sphere: there’s the ‘intrinsic’ metric, coming from thinking of it as an abstract Riemannian manifold and there’s the ‘Euclidean subspace’ metric, coming from thinking of it as a subset of $\mathbb{R}^{n+1}$. On the Earth, the intrinsic distance between Sheffield and Riverside is how far I would have to travel over the surface of the Earth to get from one to the other; whereas the Euclidean subspace distance is how far I would have to travel through the Earth if I dug a straight-line tunnel from one to the other. In previous work, Tom and I concentrated on spaces with the subspace metric, but here I am concerned with the intrinsic metric.
It is a very fun exercise to use Speyer’s formula to explicitly calculate the magnitude for $S_R^n$, the $n$-sphere of radius $R$ with its intrinsic metric (or at least I found it fun). This is done in the paper and gives the following answer.
$\left|S^n_R\right|=\begin{cases} \frac{2 \Big(\Big(\frac{R}{n-1}\Big)^2 + 1\Big)\Big(\Big(\frac{R}{n-3}\Big)^2 + 1\Big) \dots \Bigl(\Bigl(\frac{R}{1}\Bigr)^2 + 1\Bigr) }{1+e^{-\pi R}}& \text{for n even}\\ \frac{\pi R \Bigl(\Bigl(\frac{R}{n-1}\Bigr)^2 + 1\Bigr)\Bigl(\Bigl(\frac{R}{n-3}\Bigr)^2 + 1\Bigr) \dots \Bigl(\Bigl(\frac{R}{2}\Bigr)^2 + 1\Bigr) }{1-e^{-\pi R}}\quad& \text{for n odd}. \end{cases}$
The denominator rapidly tends to one as $R$ increases. So the large, or asymptotic, behaviour of the magnitude is governed by the numerator which, for fixed dimension $n$, is a polynomial in $R$ with some nice properties. For instance the leading order and constant terms can be identified in terms of classical invariants, namely the volume and the Euler characteristic, so, writing $\omega_n$ for the volume of the unit $n$-ball, the numerator is the following form:
$\tfrac{1}{n!\omega_n}vol(S^n_1)R^n +\dots+\chi(S^n_1).$
This is the kind of thing that Tom and I had been expecting for spaces with the Euclidean subspace metric. However, we will see below that the other term we can identify differs from what we naively expect in the Euclidean metric case.
One obvious question to ask here is the following.
- What is the magnitude of the $n$-sphere with the subspace metric?
…Surfaces…
All closed, compact surfaces (spheres, tori, etc.) can be equipped with a homogeneous Riemannian metric. For the sphere this should be clear as it is naturally thought of as a homogeneous space. Also the torus is commonly thought of as the quotient of the plane by an integer lattice, making it homogeneous. Similarly every higher genus surface can be obtained by quotienting the hyperbolic plane with an appropriate group of isometries, analogous to $\mathbb{Z}^2$.
For a surface with a homogeneous metric we can use Speyer’s formula from above to calculate the magnitude. I haven’t been able to find any nice closed form for the magnitude of higher genus surfaces as in the case of spheres, but there’s a nice computation giving the asymptotic behaviour. For a metric space $X$ and $t\gt0$ we can write $t X$ for $X$ with the metric scaled up by a factor of $t$. The for a surface $\Sigma$ with a homogeneous metric, as $t\to \infty$
$\begin{aligned}\left|t\Sigma\right|&= \tfrac{1}{2\pi}Area(t\Sigma)+\chi(t\Sigma) +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}). \end{aligned}$
This is precisely the kind of behaviour Tom and I had conjectured for subspaces of Euclidean spaces.
There are four main ingredients in my proof of this result: one is Speyer’s formula for the magnitude of a homogeneous metric space mentioned above, another is Watson’s Lemma, a staple of asymptotic analyis, and the other two are properties of Gauss curvature mentioned in my post on intrinsic volumes, namely that the Gauss curvature determines the circumference of small circles (at least to low order), and that the integral of the Gauss curvature is proportional to the Euler characteristic.
If we are calculating $\left|t\Sigma\right|$ using Speyer’s formula, then the key thing is to understand the denominator; so for any fixed $x_0\in \Sigma$ we want to consider
$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x.$
If $t$ is very large then because of the negative exponential, parts of $\Sigma$ which are not very close to $x_0$ will only give exponentially small contributions. This rigorous way to state this is Watson’s lemma, but the point is that for large $t$ the dominant contributions will come from local information, i.e., curvature type information.
Writing $F_r$ for the subset of $\Sigma$ at a distance of $r$ from $x_0$ we can rewrite the the above integral as follows
$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=\int_{r\ge 0}e^{-t r} Length(F_r) d r.$
But we know that for small $r$ the length, or circumference, of $F_r$ is determined at low order by the Gauss curvature $K(x_0)$.
$Length(F_r)=2\pi r \big(1 - \tfrac{1}{6}K(x_0)r^{2}+O(r^{4})\big)\qquad{as} r\to 0.$
Using this, together with Watson’s Lemma we get that the integral has the form
$\int_{x\in\Sigma} e^{-t d(x,x_0)}d x=2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big) \qquad{as} t\to\infty.$
This means that as $t\to \infty$ we find, as we wanted, that
$\begin{aligned} \left|t\Sigma\right| &=\frac{\int_{x\in\Sigma} d x}{\int_{x\in\Sigma} e^{-t d(x,x_0)}d x} =\frac{\int_{x\in\Sigma} d x}{2\pi t^{-2}\big(1 - K(x_0)t^{-2}+O(t^{-4})\big)}\\ &=\tfrac{1}{2\pi}t^2\int_{x\in\Sigma} d x + \tfrac{1}{2\pi}\int_{x\in\Sigma} K(x_0)d x +O(t^{-2})\\ &=\tfrac{1}{2\pi}Area(\Sigma)t^2+\chi(\Sigma) +O(t^{-2}), \end{aligned}$
by the fact that the Gauss curvature is constant on a homogeneous space and that the integral of the Gauss curvature is proportional to the Euler characteristic.
…and Other Homogeneous Spaces
For an arbitrary homogeneous Riemannian manifold $X$ of dimension $n$, you can use the same technique as for surfaces to calculate the leading order terms of the asymptotics of the magnitude. The key point is to identify the relevant analogues of the Gauss curvature and the Euler characteristic. These turn out to be the scalar curvature and the total scalar curvature.
At every point $x$ of a Riemannian manifold $X$ there is a scalar curvature $\tau(x)$ which specializes to twice the Gauss curvature in the case of a surface. This has the property of the Gauss curvature that at a point $x$ the volume of the set of points $F_r$ a distance $r$ from $x$ is, for small $r$, determined to leading terms by the scalar curvature:
$vol_{n-1}(F_r)= r^{n-1}\sigma_{n-1}\left(1-\tfrac{\tau(x)r^2}{6n}+\dots\right)\quad \text{as} r\to 0.$
The total scalar curvature, $tsc(X)$, is defined to be the integral of the scalar curvature;
$tsc(X)\coloneqq\int_X \tau(x) d x,$
The total scalar curvature seems to crop up in various parts of mathematics and physics, but I don’t know much about that. The connection with intrinsic volumes is that the total scalar curvature is, for closed manifolds, the next non-trivial intrinsic volume after the volume:
$\mu_{n-2}(X) = \frac{1}{4\pi}tsc(X).$
In the case that $X$ is a surface (so $n=2$) we recover the relationship between the total Gauss curvature and the Euler characteristic
Running through through the same analysis as for the case of surfaces above, you find that for $X$ a homogeneous Riemannian manifold the magnitude looks asymptotically as follows:
$\left|t X\right| = \tfrac{1}{n!\omega_n}vol(X)t^n+\tfrac{n+1}{6n!\omega_n}tsc(X)t^{n-2}+O(t^{n-4}).$
There are a few questions that jump out at this point.
- Can these methods be used to calculate further terms in the asymptotic expansion for homogeneous manifolds?
- Are these leading terms the same for the asymptotic behaviour of non-homogeneous Riemannian manifolds?
I should probably note here that the asymptotic behaviour is only one part of the picture and it would also be good to know what is going on for ‘small’ spaces: what is the magnitude measuring?
Re: On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces
Thanks for the post and paper. I’m a big fan of this magnitude of metric spaces stuff.
Once every couple of years I pick up Street’s paper “Algebra of oriented simplexes” hoping my math kung fu has progressed enough for me to finally understand it. Coincidentally, the last time I picked it up recently made me think of this magnitude of metric spaces stuff. Probably for no good reason other than lack of sleep though. Let me throw some thoughts out there and see if anything makes sense…
I liked the idea of relating the 1-cocycle condition to a composite triangle in a category, i.e.
$\partial[012] = [12]-[02]+[01] = 0$
rearranges to
$[02] = [01]+[12]$
which is interpreted as a commuting triangle.
Similarly
$\partial[0123] = [123]-[023]+[013]-[012] = 0$
rearranges to
$[123]+[013] = [023]+[012].$
The pictures that go with this are great.
Then I had a whacky idea. What if we turn the square brackets into magnitudes instead, i.e.
$|02| = |01|+|12|.$
This says the magnitude of [02] equals the sum of the magnitudes of [01] and [12]. But if magnitude is “volume” then we are talking about lengths of the sides of a triangle. When is the sum of two lengths of a triangle equal to the length of the third side? When the triangle is degenerate, i.e. has no area. Area is zero? Boundary is zero? Could they be related?
This idea bumps up trivially in dimension.
The 2-cocycle condition gives
$|123|+|013| = |023|+|012|$
which says the sum of areas of two triangles of a tetrahedron are equal to the sum of the areas of the two remaining triangles of the tetrahedron. When does that happen? It happens when the 3-simplex is degenerate, i.e. it has no volume.
The picture is kind of pretty. You partition the boundary of an $n$-simplex into two $(n-1)$-complexes. The $(n-1)$-cocycle condition says the magnitude of these two complexes are equal.
We have magnitudes of metric spaces and magnitudes of categories, is there a such thing as a magnitude of a diagram in a category?
Apologies if this makes absolutely no sense and is completely irrelevant to this magnitude business :)