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March 12, 2010

Intrinsic Volumes and Weyl’s Tube Formula

Posted by Simon Willerton

I would like to start to explain something about the “intrinsic volumes” for Riemannian manifolds, which are some fundamental differential geometric invariants including the volume, the total scalar curvature and the Euler characteristic, and they go by various names including curvature measures, Killing-Lipshitz curvatures, generalized curvatures and quermassintegrale.

The reason I’m currently interested in these is that I have just about finished a paper on the magnitude of some homogeneous Riemannian manifolds and these intrinsic volumes or generalized curvatures seem to be related to the asymptotics of the magnitude.

You may well remember that the magnitude (or cardinality) of a metric space was introduced by Tom Leinster in a post here at the Café as a special case of the Euler characteristic of an enriched category. The fact that the asymptotic behaviour of the magnitude is related to intrinsic volumes for polyconvex sets has already been mentioned here and here. However, the paper I’m finishing has to do with smooth things rather than polyconvex ones. I will mention connections with polyconvex sets at the end of this post.

Here I will restrict myself to the basic case of closed surfaces in 33-space, although everything generalizes, and use as motivation the celebrated Tube Formula of Weyl, which you might wish to ponder before reading on.

Tube Problem. Suppose Σ\Sigma is a surface in 3\mathbb{R}^3; thicken it up to form the ϵ\epsilon-neighbourhood N ϵΣN_\epsilon \Sigma consisting of all points within ϵ\epsilon of Σ\Sigma. Supposing that ϵ\epsilon is sufficiently small, how does the volume of N ϵΣN_\epsilon \Sigma depend on ϵ\epsilon and Σ\Sigma?

Very little knowledge of differential geometry will be assumed.

Curvatures and the Tube Formula for surfaces in 3\mathbb{R}^3

I will start with the very basics of the notion of curvature of curves and surfaces, then get on to explaining Weyl’s Tube formula. All of this generalizes to arbitrary submanifolds of arbitrary Euclidean spaces, but I’ll stick with this most basic example today.

My standard references for this are the following books:

Both are extremely interesting, but both I find flawed in some way, and both are quite notation heavy. Anyway, let’s get on with the story.

Suppose we have a surface Σ\Sigma smoothly embedded in 3\mathbb{R}^3 we want to define some notion of curvature at each point.

What we start with is knowing how to define the curvature of a planar curve, namely if we have a point on a curve then there is a circle in 2\mathbb{R}^2 (perhaps of infinite radius!) which best approximates the curve at that point, this is called the osculating circle at that point. Clearly the smaller the circle, the more curved the curve is at that point, and the curvature at that point is defined to be the reciprocal of the radius this osculating, or best-approximating, circle.

Now back to our surface. If we have a point xx on the surface Σ\Sigma then there is a well defined normal line to xx, and we can decide that one of the two normal directions is going to be positive. Picking a direction on the surface at xx will give us a plane spanned by that direction and the normal. The intersection of the surface with this plane is locally a curve and the curvature of Σ\Sigma at xx in the given direction is defined to be the curvature of this curve. Here is a picture of the osculating circle corresponding to a direction on a surface.

Osculating circle on a surface

The black arrows are the chosen positive normal and the direction we are interested in. The orange curve is a curve in the surface which is tangent to our specified direction and the red circle is the osculating circle, or circle which best fits the curve. So the curvature of the surface in the shown direction is the reciprocal of radius of the red circle.

We will give the curvature a positive sign if the osculating circle is on the opposite side to the positive normal direction and negative otherwise. Thus in the above picture the given direction has negative curvature, but as the direction is changed at the indicated point, the curvature will become positive. So for each direction in the surface at xx we have a signed curvature. The principal curvatures, κ 1\kappa_1 and κ 2\kappa_2, at xx are taken to be the maximum and minimum values of the curvature. From these we can define the following two invariants at xx:

  • Gauss curvature K(x):=κ 1κ 2K(x):=\kappa_1\kappa_2;
  • mean curvature H(x):=(κ 1+κ 2)/2H(x):=(\kappa_1+\kappa_2)/2.

Although these look to be of the same nature, they have quite different properties. At a basic level you can see that if you switch the chosen normal direction then the principal curvatures will both switch signs, so Gauss curvature is independent of the choice of normal direction, but the sign of the mean curvature depends on the choice of normal direction. More generally, the Gauss curvature is intrinsic meaning that if you are an ant living on the surface then you can find out the Gauss curvature by just making measurement on the surface, whereas the mean curvature is not intrinsic.

We will see why the latter is true in a minute, but first I will tell you how to measure the Gauss curvature of the Earth on which you are living. The idea is that you measure the circumference of the circle consisting of all points on the Earth a distance rr away from a fixed point. If you were on a plane then the circle would be of length 2πr2\pi r, but because the Earth is (approximately) spherical the circle will have a smaller circumference than that. For small rr the deficit in the circumference is to first approximation given by the Gauss curvature:

Circumference=2πr(1K(x)6r 2+O(r 4)).Circumference = 2\pi r\left( 1- \frac{K(x)}{6}r^2+O(r^4)\right).

and this determines the Gauss curvature at that point. For instance, it is an exercise in high-school geometry to show that on a radius RR sphere the circumference of a radius rr circle is

2πRsinrR=2πr(116R 2r 2+O(r 4))2\pi R sin\tfrac{r}{R}=2\pi r \left(1- \tfrac{1}{6R^2}r^2+O(r^4)\right)

so the Gauss curvature of the radius RR sphere is 1/R 21/R^2 at every point. This should not be surprising as you can see straight away that the principal curvatures at every point are both 1/R1/R (the osculating circle is always a great circle on the sphere).

I could have equally well talked about the deficit in the area of a disc of radius rr and expressed that to first approximation in terms of the Gauss curvature. The moral of this is that if you have a goat tied to a post on a fixed length rope then the more positive the Gauss curvature of the field is, the less grass the goat will have to eat!

The fact that the Gauss curvature is intrinsic to the surface is known as Gauss’ Theorema Egregium, or Remarkable Theorem.

I should now point out why the mean curvature is not intrinsic. Consider a piece of paper. If the piece of paper is flat then the principal curvatures are all zero, so the mean curvature is zero. If you roll the piece of paper into a cylinder of radius rr then at each point the principal curvatures are 00 and 1/r1/r so the mean curvature is 1/2r1/2r. However if you were an ant living on the piece of paper then none of your intrinsic measurements would be able to distinguish the two situations – the lengths and angles of things are not changed. Thus the mean curvature is not an intrinsic invariant. As the Gauss curvature is an intrinsic invariant it ought to be the same in both cases, and indeed it is just zero each time.

The next remarkable thing about the Gauss curvature is the Gauss-Bonnet Formula which says that if you have a closed surface then when you integrate Gauss curvature then you obtain, up to scale, one of the most fundamental invariants of geometry and topology – the Euler characteristic – remember that for an orientable surface the Euler characteristic is just related to the “number of holes” or genus g(Σ)g(\Sigma) by χ(Σ)=22g\chi(\Sigma)=2-2g.

Gauss-Bonnet Formula. For an oriented closed surface Σ\Sigma in 3\mathbb{R}^3 the integral of the Gauss curvature is proportional to the Euler characteristic:

ΣK(x)dx=2πχ(Σ).\int_\Sigma K(x) dx = 2\pi\chi(\Sigma).

So, for instance, no matter how you embed a sphere in Euclidean the integral of the Gauss curvature is always going to be 4π4\pi: if you make it more positively curved at one point you have to compensate by making it less curved at another point. Similarly if you embed a torus in 3\mathbb{R}^3 then the integral of the Gauss curvature is going to be 00 so it will have to be positively curved in some places and negatively curved in others.

Now I want to get to the Tube Formula which is a general form is due to Weyl, but the following version for surfaces was undoubtably known before him. The set up is the following. Suppose you have a surface Σ\Sigma embedded in 3\mathbb{R}^3 and we define N ϵΣN_\epsilon\Sigma to be the ϵ\epsilon-neighbourhood of Σ\Sigma, that is all of the points within an ϵ\epsilon distance of Σ\Sigma. What is the the volume of N ϵΣN_\epsilon\Sigma? The answer is perhaps a little surprising the first time you see it.

Weyl’s Tube Formula for a Surface in 3\mathbb{R}^3. For an oriented closed surface Σ\Sigma in 3\mathbb{R}^3, and for ϵ\epsilon sufficiently small, the volume of the ϵ\epsilon-neighbourhood is given by the following formula:

vol(N ϵΣ)=2Area(Σ)ϵ+4π3χ(Σ)ϵ 3.vol(N_\epsilon \Sigma)= 2 Area(\Sigma) \epsilon + \frac{4\pi}{3}\chi(\Sigma)\epsilon^3.

For the standard embedding of a radius RR sphere this is a straight forward calculation as the ϵ\epsilon–neighbourhood is just a radius (R+ϵ)(R+\epsilon) 3–ball with a 33-ball of radius (Rϵ)(R-\epsilon) removed (so “ϵ\epsilon sufficiently small” here means ϵ<R\epsilon\lt R). For a general surface the proof is a one page differential geometry calculation which can be found in the book by Gray. I should point out, to remove any mystery, that the coefficients 22 and 4π/34\pi/3 are the volumes of the unit 11–ball and unit 33–ball respectively.

The first thing to notice about this is that it is a polynomial in ϵ\epsilon. The next thing to notice is that the dependency on Σ\Sigma is via integrals of local things on the surface, namely the area, Σdx\int_\Sigma dx, and the Euler characteristic, Σκ 1κ 2/2πdx\int_\Sigma \kappa_1\kappa_2/2\pi dx. The surprising thing is that the expression only depends on intrinsic properties of the surface – it does not depend on the embedding per se.

The analogous thing one dimension down is that if you have a smooth, simple, closed curve Γ\Gamma in the plane then the area of the ϵ\epsilon–neighbourhood is just

Area(N ϵΓ)=2Length(Γ)ϵArea(N_\epsilon \Gamma)= 2 Length(\Gamma) \epsilon

which is independent of the embedding. However, if you just took the inside piece, or the outside piece of the neighbourhood, then you would have got something which would have depended on the curvature of the embedding, but when you add the two bits together then this dependency cancels out.

A similar thing happens with the inside and the outside of the ϵ\epsilon-neighbourhood of a surface. The contributions which cancel out in that case actually come from the mean curvature – remember we observed that the mean curvature switches sign if you switch your chosen normal direction, well this is related to that phenomenon. [In fact in higher dimensions, to get the tube formula you need to integrate certain curvatures over the unit normal bundle, and the ‘odd’ ones will cancel themselves out, as has happened with the mean curvature here.]

One way to write the Tube Formula for a Surface which will generalize is to write ω i\omega_i for the volume of the unit ii–ball, and write μ 2\mu_2 and μ 0\mu_0 for the area and Euler characteristic respectively, and taking μ 3\mu_3 and μ 1\mu_1 to be zero, then for a 22–dimensional surface in 33–dimensional space and ϵ\epsilon sufficiently small:

vol(N ϵΣ)=μ 2(Σ)ω 1ϵ+μ 0(Σ)ω 3ϵ 3= i=0 3μ 3i(Σ)ω iϵ i.vol(N_\epsilon \Sigma)= \mu_2(\Sigma) \omega_1\epsilon + \mu_0(\Sigma)\omega_3\epsilon^3= \sum_{i=0}^3\mu_{3-i}(\Sigma) \omega_i\epsilon^i.

For arbitrary submanifolds of arbitrary Euclidean spaces you get appropriately generalized Lipschitz-Killing curvatures {μ i}\{\mu_i\} and a Tube Formula of the above form. I hope to talk about these another time. However, I just want to briefly try to tie this in with intrinsic volumes as described by Tom previously.

Intrinsic volumes and polyconvex sets

Back in his original post on the magnitude of metric spaces (then called the cardinality of metric spaces), Tom described the intrinsic volumes for certain basic subsets of Euclidean space – the finite unions of convex sets, also known as polyconvex sets. For a fixed a Euclidean space k\mathbb{R}^k the invariant valuations (or measures in Tom’s terminology) are defined to be the the functions on the set of polyconvex sets which are invariant under rigid motions, which are continuous in an appropriate sense and which are finitely additive, in other words such a function μ\mu satisfies the inclusion-exclusion principle:

μ(AB)=μ(A)+μ(B)μ(AB)forpolyconvexAandB.\mu(A\cup B)=\mu(A)+\mu(B)-\mu(A \cap B) \qquad for polyconvex A and B.

Here are some standard examples:

μ k =k-dimensional volume μ k1 =12"surface area" μ 0 =Euler characteristic \begin{aligned} \mu_k &= k\text{-dimensional volume}\\ \mu_{k-1} &= \tfrac{1}{2}\text{"surface area"}\\ \mu_{0}&= \text{Euler characteristic} \end{aligned}

Hadwiger’s Theorem says that these three invariants can be extended to a basis μ 0,μ 1,,μ k\mu_0, \mu_1, \dots, \mu_k of all the invariant valuations. The valuation μ i\mu_i is called the iith intrinsic volumes and is homogeneous of degree ii in the sense that as a subset is scaled up by a factor of t>0t\gt 0 then the μ i\mu_i scales up by a factor of t it^i – think about the examples above – so μ i(tA)=t iμ i(A)\mu_i(t A)=t^i \mu_i(A), where tAt A means AA scaled up by a factor of tt.

One of the most well-known results involving intrinsic volumes is Steiner’s Formula.

Steiner’s Formula. Suppose that KK is a convex subset in k\mathbb{R}^k, that N ϵKN_\epsilon K is the set of all points within ϵ\epsilon of KK and that ω i\omega_i is the volume of the unit ii-ball, then we have the following formula for the volume of the ϵ\epsilon-neighbourhood of KK: vol(N ϵK)= i=0 kμ ki(K)ω iϵ i.vol(N_\epsilon K) = \sum_{i=0}^k\mu_{k-i}(K)\omega_i \epsilon^i.

If you have not seen this before then you should draw a picture of the ϵ\epsilon-neighbourhood of a (convex!) triangle in the plane and identify the three terms on the right hand side in the expression for the area of the neighbourhood; namely something involving the Euler characteristic of the triangle(which is 11!), something involving (half) the perimeter of the triangle, and something involving the area of the triangle.

Steiner’s Formula looks remarkably similar to Weyl’s Tube Formula, but is definitely different. The subspaces that are being considered namely convex sets and closed submanifolds are quite distinct types of space – convex sets are flat with possibly curved, possibly non-smooth boundary; closed submanifolds are curved, smooth and have no boundary. However, there is, of course, a common generalization of these two theorems which I leave you to ponder.

The point is that the intrinsic volumes are actually defined on a considerably greater class of subsets of k\mathbb{R}^k which includes both polyconvex sets and submanifolds. This was originally done by Federer in his work on curvature measures but generalized by others since. The book of Morvan goes into details about the history. The relevant fact about the subsets is that they need to have some appropriate notion of unit normal bundle, known as a normal cycle. Moreover, these invariants do seem to crop up in lots of places and be of a fundamental nature.

Of course it remains to be seen quite how much of these invariants are seen by the magnitude of metric spaces.

Posted at March 12, 2010 6:13 PM UTC

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8 Comments & 4 Trackbacks

Re: Intrinsic Volumes and Weyl’s Tube Forumla

Erin Pearse did a bunch of work relating the tube formula for fractals like the Koch curve to the “complex dimensions” of the fractal, which are the poles of the fractal’s zeta function.

Posted by: Mike Stay on March 12, 2010 10:08 PM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Forumla

Looks like Erin is currently at U. Iowa.

Posted by: Mike Stay on March 12, 2010 10:10 PM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Forumla

Yes I did look at that a little a while ago. In mine and Tom’s asymptotic paper we looked at the asymptotics of the magnitude of the Cantor set and in my computational paper I did some calculations for the magnitude of the Sierpinski gasket; so I was looking around to see what people had done about intrinsic volumes of fractals. Unfortunately, from a naive flicking through I couldn’t relate their approach with the kind of results we got.

For instance for the ternary Cantor set of length tt, as tt\to \infty the magnitude goes like

f(t)t log 32,f(t)t^{\log_3{2}},

where f(t)f(t) satisfies f(3t)=f(t)f(3 t)= f(t) and it would seem to be like some intrinsic volume of homogoneity log 32{\log_3{2}} – this number being the Hausdorff dimension of the Cantor set. Explicity, f(t)f(t) is the following:

f(t)t log 3(2)12 i= 2 itanh(t23 i).f(t)\coloneqq t^{-\log_3(2)}\frac{1}{2}\sum_{i=-\infty}^{\infty} 2^i\tanh\left(\frac{t}{2\cdot 3^i}\right).

Posted by: Simon Willerton on March 13, 2010 5:00 AM | Permalink | Reply to this

explanatory pictures for Hadwiger’s theorem; Re: Intrinsic Volumes and Weyl’s Tube Forumla

A Neighborhood of Infinity, Sunday, August 13, 2006

What can we measure? Part I

“I recently mentioned this course on Discrete Differential Geometry. One section, whose title I have shamelessly stolen, discusses a curious result known as Hadwiger’s theorem. What it doesn’t include, however, are explanatory pictures. So read on for a an elementary pictorial introduction to Hadwiger’s theorem….”

At the bottom of his page he adds:
Update: Baez talks about some of this stuff here where it’s related to Schanuel’s characterisation of the Euler characteristic. Yet again I can’t write about anything without JB haven’t beaten me to it. I could swear that guy’s following me around… :-)

Posted by: Jonathan Vos Post on March 13, 2010 3:50 AM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Formula

This is just a quick thought about Gauss curvature vs mean curvature. There’s an old notion of “generalized average”, which goes back to (at least) the book Inequalities of Hardy, Littlewood and Polya. Let p 1,,p np_1, \ldots, p_n be non-negative numbers summing to 1. Let x 1,,x nx_1, \ldots, x_n be real numbers. For each tt in \mathbb{R}, let M t=(p ix i t) 1/t. M_t = (\sum p_i x_i^t)^{1/t}. Then M tM_t is a kind of average of x 1,,x nx_1, \ldots, x_n, weighted by p 1,,p np_1, \ldots, p_n. The expression isn’t defined at t=0t = 0, but it does have a limit there, namely M 0=x i p i. M_0 = \prod x_i^{p_i}. When p 1==p n=1/np_1 = \cdots = p_n = 1/n, we have:

  • M 1=arithmeticmeanM_1 = arithmetic mean,
  • M 0=geometricmeanM_0 = geometric mean,
  • M 1=harmonicmeanM_{-1} = harmonic mean.

It’s not hard to show (for arbitrary p ip_i) that M tM_t is increasing in tt. This implies, for instance, the AM-GM inequality.

(I guess M tM_t is sometimes undefined, e.g. if x i=0x_i = 0 and t<0t &lt; 0. I don’t know what one says in that case.)

Now, this continuum of means is closely related to the continuum of entropies (and the continuum of diversity measures) that I’ve been thinking about. Shannon entropy corresponds to M 0M_0. Shannon entropy is also special among all kinds of entropy: it has properties that nothing else has. So this is a reason to expect M 0M_0 to be special in all situations.

You mention two kinds of curvature: Gauss, which is the square of the geometric mean (M 0M_0), and mean curvature, which is the arithmetic mean (M 1M_1). Let’s say that the tt-curvature is M t(κ 1,κ 2)M_t(\kappa_1, \kappa_2). You show that while the 0-curvature is intrinsic, the 1-curvature is not. The example you use to show that it’s not also shows that tt-curvature is not intrinsic either whenever t0t \neq 0. So in this sense too, M 0M_0 is special among all the M tM_t’s.

Posted by: Tom Leinster on March 21, 2010 12:34 AM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Formula

Our library doesn’t have Gray’s book Tubes. Moreover, if I search for “Gray, Tubes”, the library’s search engine replies:

Did you mean: gray mullets?

Posted by: Tom Leinster on March 21, 2010 3:25 AM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Formula

Which sort of mullet does the recommendation refer to:

Posted by: David Roberts on March 22, 2010 12:44 AM | Permalink | Reply to this

Re: Intrinsic Volumes and Weyl’s Tube Formula

I don’t know for sure, but the one book under “gray mullets” is in the aquaculture section. Your guess is as good as mine…

Posted by: Tom Leinster on March 22, 2010 2:21 AM | Permalink | Reply to this
Read the post On the Magnitude of Spheres, Surfaces and Other Homogeneous Spaces
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Read the post Mixed Volume
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