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On the Kinematic Formula in the Lives of the Saints Danny Calegari

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On the Kinematic Formula in the Lives of the Saints Danny Calegari SHORT STORIES On the Kinematic Formula in the Lives of the Saints Danny Calegari Saint Sebastian was an early Christian martyr. He served as a captain in the Praetorian Guard under Diocletian un- til his religious faith was discovered, at which point he was taken to a field, bound to a stake, and shot by archers “till he was as full of arrows as an urchin1 is full of pricks2.” Rather miraculously, he made a full recovery, but was later executed anyway for insulting the emperor. The trans- pierced saint became a popular subject for Renaissance painters, e.g., Figure 1. The arrows in Mantegna’s painting have apparently ar- rived from all directions, though they are conspicuously grouped around the legs and groin, almost completely missing the thorax. Intuitively, we should expect more ar- rows in the parts of the body that present a bigger cross section. This intuition is formalized by the claim that a subset of the surface of Saint Sebastian has an area pro- portional to its expected number of intersections with a random line (i.e., arrow). Since both area and expectation are additive, we may reduce the claim (by polygonal ap- proximation and limit) to the case of a flat triangular Saint Sebastian, in which case it is obvious. Danny Calegari is a professor of mathematics at the University of Chicago. His Figure 1. Andrea Mantegna’s painting of Saint Sebastian. email address is [email protected]. 1 hedgehog This is a 3-dimensional version of the classical Crofton 2quills formula, which says that the length of a plane curve is pro- For permission to reprint this article, please contact: portional to its expected number of intersections with a [email protected]. random line. A famous application is known as “Buffon’s DOI: https://doi.org/10.1090/noti2117 needle”. In the Euclidean plane we consider the family of 1042 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 7 Short Stories all horizontal lines whose 푦 coordinate is an integer, and dimension). Chern generalizes this as follows: for every we drop a needle of length 1 so that it lands somewhere even number 푒 ≤ 푝 + 푞 − 푛 he proves there is a formula at random. Then the probability that the needle crosses one of the lines is 2/휋 (i.e., the average of | sin(휃)|, where ∫ 휇푒(푀 ∩ 푔푁) 푑푔 = ∑ 푐푖휇푖(푀)휇푒−푖(푁) 휃 is the angle the needle makes with the horizontal). In 푔∈퐺 even 푖≤푒 fact, the only relevance of the straightness of the needle is where 푐푖 are universal constants depending only on 1 that a straight needle of length can cross at most one of 푛, 푝, 푞, 푒, and the 휇푒 are certain geometric invariants of 푀 the horizontal lines in at most one point. If you trade in and 푁 concentrated in “codimension 푒” (so to speak). Fur- your needle for a stick of spaghetti of the same length (i.e., thermore, Chern’s formula is valid in the “noodly” sense “Buffon’s noodle”) and then cook the spaghetti and drop that the invariants 휇푖 depend only on the intrinsic Rie- it at random in the plane, the expected number of cross- mannian geometry of the manifolds, and not on how they ings with the horizontal lines is still 2/휋, no matter what are isometrically embedded in Euclidean space. complicated physical process determines its shape. The geometric invariants 휇푒 appear in Weyl’s (closely re- lated) formula for the volume of a (sufficiently narrow) tube of radius 휌 about a 푘-dimensional submanifold 푋 of ℝ푛. Let 푇(푋, 휌) denote the tube around 푋 of radius 휌, where 휌 is sufficiently small depending on 푋. Then vol(푇(푋, 휌)) (푒 − 1)(푒 − 3) ⋯ 1 = 푂 ∑ 휇 (푋)휌푛−푘+푒, 푛−푘 (푛 − 푘 + 푒)(푛 − 푘 + 푒 − 2) ⋯ (푛 − 푘) 푒 even 푒≤푘 푖 where 푂푖 is the volume of the unit sphere in Euclidean ℝ , Figure 2. A random noodle of length 1 has 2/휋 expected and the 휇푒(푋) are certain integrals over 푋 of invariant poly- crossings with the horizontal lines. nomials in the matrix entries of the second fundamental form. Evidently the leading order term for small 휌 is just 푛−푘 Crofton’s formula can be greatly generalized. For ex- 푂푛−푘vol푘(푋)휌 ; i.e., 휇0(푋) = vol푘(푋). One may think ample, Kang and Tasaki give a formula for the expected of Weyl’s formula as a special case of (Federer’s version of) 푛 number of intersections of a (real) surface 푆 in ℂℙ with a the kinematic formula, taking 푀 = 푋 and 푁 equal to a ball random hyperplane. The number of such intersections is 퐵휌 of small radius 휌. Then 휇푖(퐵휌) is (up to a constant) just 1 the appropriate power of 휌. ∫1 + cos2(휃 ) 푑area(푥), 2휋 푥 Weyl’s formula is exact, but only valid for 휌 small 푆 enough so that the exponential map is an embedding on 푥 ∈ 푆 휃 where for a point the “Kähler angle” 푥 measures the normal disk bundle of radius 휌. What the formula re- 푇푆 푣 , 푣 how far is from being complex: if 1 2 is an orthonor- ally computes is the integral over this normal disk bun- 푇 푆 cos(휃 ) 푖푣 mal basis for 푥 , then 푥 is the inner product of 1 dle of the pullback of the Euclidean volume form under 푣 푆 with 2. For example, if is a holomorphic curve, then the exponential map. For larger 휌 one must compute vol- 휃 = 0 and the area of 푆 is 휋 times the degree, whereas for 푆 2 ume with multiplicity (where different sheets of the image equal to the “standardly embedded” ℝℙ one has 휃 = 휋/2 of the exponential map overlap each other) and with sign and the area is 2휋. (where the Jacobian of the exponential map has negative One of the most beautiful generalizations is the so- sign). For accuracy we should call this the algebraic volume called kinematic formula of Chern (actually, a more general of 푇(푋, 휌), to distinguish it from the naive volume. Fig- formula still, valid for manifolds with boundary among ure 3 shows a region where three local sheets of the expo- other things, was obtained seven years earlier by Federer). nential map overlap, two with positive sign and one nega- “Kinematic” here refers to the group 퐺 of isometries of tive. 푝 푞 Euclidean space. Given submanifolds 푀 and 푁 of Eu- One can think of Weyl’s formula as a Taylor series, but 푛 clidean ℝ , a modest improvement of Crofton’s formula if so it has two properties that are rather startling at first says glance: 1. the nonzero terms all have the same parity (equal to ∫ vol푝+푞−푛(푀 ∩ 푔푁) 푑푔 = 푐 vol푝(푀)vol푞(푁), 푔∈퐺 the codimension of 푋); and 2. there are only finitely many nonzero terms! where vol푖 denotes 푖-dimensional volume, and 푐 is a univer- sal constant depending only on the dimensions 푛, 푝, 푞, and Actually a little reflection makes these properties appear the normalization of the measure on 퐺 (the actual compu- less preposterous. Firstly, if we express 푇(푋, 휌) as the inte- tation of 푐 was first carried out by Santal´o,who showed it gral of a density 푇(푋, 휌, 푣) over vectors 푣 in the unit normal is a ratio of products of Euclidean unit spheres of various bundle 휈, then if we expand 푇(푋, 휌, 푣) for any fixed 푣 as a AUGUST 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1043 Short Stories by 푁휌(푥, ±1) = exp(±휌푣), where 푣 is the positive unit nor- mal to 푋 at 푥. For big 휌, it is approximately true that 푁휌 maps 푋 × 1 (resp., 푋 × −1) to the sphere of radius 휌 by 휌 ⋅ 퐺 (resp., −휌 ⋅ 퐺) where 퐺 is the Gauss map. If 푘 is even the antipodal map has degree −1. Since the components of 푋 × ±1 have opposite orientations, it follows that the oriented degree of both maps is equal to deg(퐺) = 휒(푋)/2. 푛 Thus the coefficient of 휌 in Weyl’s formula is 푂푛휒(푋), and by running this argument in reverse, we obtain a proof of the Chern–Gauss–Bonnet formula! Taking 푒 = 푝 + 푞 − 푛 in Chern’s formula one has the rather extraordinary conclusion that if one drops 푝- and 푞-dimensional noodles 푀 and 푁 randomly into ℝ푛 the expected Euler characteristic of their intersection depends only on the intrinsic Riemannian geometry of 푀 and 푁. Saints alive! Figure 3. A ‘tube’ of sufficiently large radius has volume which must be counted with multiplicity and with sign. In this figure, the red region is counted with multiplicity 1, the green AUTHOR’S NOTE. Chern’s formula is proved in the pa- region with multiplicity 1 + 1 − 1, and the blue region with per “On the Kinematic Formula in Integral Geometry,” multiplicity 1 + 1. J. Math. Mech. 16 (1966), no. 1, 101–118. Federer’s pa- per “Curvature measures,” Trans. AMS 93 (1959), 418– power series in 휌 we will have 푇(푋, 휌, −푣) = 푇(푋, −휌, 푣). 491, is also recommended. Thus only the terms with an even exponent will survive after integrating over 푣. Secondly, if we approximate 푋 by a polyhedron 푌, it is obvious that each codimension 푒 simplex 휎 contributes an 푛−푘+푒 expression of the form 푐푒 훼(휎)vol푘−푒(휎)휌 , where 푐푒 is a constant depending only on 푛, 푘, 푒, and where 훼(휎) (the angle excess) is the volume of a certain (dual) immersed spherical polyhedron whose faces are inductively associ- ated to higher-dimensional simplices 휏 incident to 휎.
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