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APPEARED IN BULLETIN OF THE

arXiv:math/9210216v1 [math.DG] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 261-265CURVATURE, TRIAMETER, AND BEYONDKarsten Grove and Steen MarkvorsenAbstract. In its most general form, the recognition problem in Riemannian geome-try asks for the identification of an unknown Riemannian manifold via measurementsof metric invariants on the manifold.

We introduce a new infinite sequence of invari-ants, the first term of which is the usual diameter, and illustrate the role of theseglobal shape invariants in a number of recognition problems.It is apparent that information about basic geometric invariants such as curva-ture, diameter, and volume alone does not suffice in order to characterize Riemann-ian manifolds in general. For this reason it is not only natural but necessary topursue and investigate other metric invariants.

Recently such investigations haveincluded Gromov’s filling radius (cf., e.g., [G, K1, K2, W]), excess invariants (cf.,e.g., [GP2, O, PZ]), and Urysohn’s intermediate diameters (cf., e.g., [U, G, K3]).The purpose here is to introduce new metric invariants and announce relatedrecognition theorems. Our main concern is an infinite sequence: diameter, triame-ter, quadrameter, quintameter,.

. .

, etc., whose nth term is based on measurementson (n + 1)-tuples of points. Precisely, if (X, dist) is any compact metric space, theq-extent, xtq X, of X is the maximal average distance between q points in X, i.e.,xtqX =max(x1,...,xq)Xi

. .

, xq ∈X.With this definition, xt2 X = diamX, xt3X = triamX, etc., and obviouslyxt2X ≥xt3X ≥· · · ≥xtqX ≥xtq+1X ≥· · · ≥xtX,where xt X = limq xtqX is called the extent of X. It is easy to see that12diamX ≤xtX < diamXfor any compact X and that these inequalities are optimal.

Somewhat surprisingly,however, it turns out that xt X is related to the excess, exc X, as defined in [GP2].Namely, if X has almost minimal extent, its excess is almost zero. From this andseveral explicit computations (cf.

[GM]), it appears that the above extent invariantsare sensitive to asymmetries of a space and, therefore, should be thought of not assize invariants, but as global shape invariants in the same way that curvatures arethought of a local shape invariants.Received by the editors November 5, 19911980 Mathematics Subject Classification (1985 Revision). Primary 53C20, 51K10, 53C23Both authors were supported in part by the Danish Research Council; the first author was alsosupported by a grant from the National Science Foundationc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2KARSTEN GROVE AND STEEN MARKVORSENSpaces with large extentsThe first step in the recognition program for a given metric invariant is to in-vestigate its range when restricted to various subclasses of metric spaces (cf. [G2,GM]).

For the individual q-extents there are trivial optimal estimates like the onegiven for xt X above, even when restricted to Riemannian manifolds. This becomesan attractive problem, however, when a lower curvature bound is present, and thuslocal and global shape invariants are balanced against one another.Specifically, for each k ∈R and R > 0, let Dnk(R) denote the closed metric R-ballin the simply connected n-dimensional complete space form Snk of constant curva-ture k. For any closed Riemannian n-manifold M, whose sectional curvature andradius satisfy sec M ≥k and rad M ≤R, standard Toponogov distance comparisonyields(∗)xtq M ≤xtq Dnk(R),for any integer q ≥2.

Recall that rad X ≤R if and only if X = D(x, R) for somex ∈X (cf. [SY, GP4]).When k > 0 and R ≥π/2√k, it turns out that xtq Dnk(R) = xtqSnk = xtq[0, π/√k]for all q, and all q-extenders, i.e., q-tuples of points realizing xtq, can be explic-itly described (cf.

[GM, N]). Moreover the inequalities (∗) are optimal in this case.The following range/recognition theorem generalizes Toponogov’s maximal diame-ter theorem (cf.

[CE]):Theorem A. Let M be a closed Riemannian n-manifold, n ≥2, with sec M ≥1.Then xtq M ≤xtq Sn1 for every q ≥2.If equality holds for some q ≥2, then M is isometric to Sn1 .For any ε > 0 and any π/2 ≤R ≤π there is a Riemannian manifold M ≃Snwith sec M ≥1, radM ≤R, and xtq M ≥xtq Sn1 −ε.Note that since diam M ≥xtqM for any q ≥2, the diameter sphere theorem[GS] implies that M is homeomorphic to Sn if xtq M > π/2 in the above theorem.In the remaining cases we only know xtq Dnk(R) when q ≤n+1 (cf.

[GM, T, H]).Here xtn+1 Dnk(R) is of particular interest because for R < π/4√k (if k > 0) thereis only one (n + 1)-extender, namely, the vertices of the unique maximal, regularlyinscribed n-simplex, ∆nk(R) in Dnk(R). Here without the loss of generality we mayassume R = 1.

If r(n, k) is the radius of the largest ball inscribed in ∆nk = ∆nk(1),then r(n, ·) : (−∞, (π/2)2) →(0, 1) is a strictly increasing continuous function foreach n ≥2. Let k(n) be determined by r(n, k(n)) = 1/2.

The optimality questionof (∗) is then resolved for q = n + 1 according to the following.Theorem B. Fix an integer n ≥2 and k < ( π4 )2.

For any closed Riemanniann-manifold M with sec M ≥k and rad M ≤1,xtn+1 M < xtn+1 Dnk(1),and this inequality is optimal if and only if k ≤k(n).There is an ε(n) > 0, so that if k = k(n) andxtn+1 M ≥xtn+1 Dnk(n)(1) −ε(n),

CURVATURE, TRIAMETER, AND BEYOND3then M is homeomorphic to Sn.In contrast to the proof of Theorem A, this result is proved using convergencetechniques. It follows from these techniques via [GPW] (cf.

also a recent result ofPerelman announced in [BGP]) that there are at most finitely many topologicaltypes of manifolds M as in Theorem B, for which xtn+1 M ≥xtn+1Dn−1k(1) + ε,for any fixed ε > 0.Once it has been observed that the double D∆nk = ∆nk` ∆nk/(∂∆nk ∼∂∆nk)of the simplex ∆nk has curvature curv D∆nk ≥k in distance comparison sense andrad D∆nk = 1 when k ≤k(n), the optimality statement in Theorem B is fairlytrivial. The hard part is to prove the recognition statement when k = k(n) andthe nonoptimality statement when k > k(n).

This, on the other hand, follows fromTheorem C below together with results and tools developed in [GP3, GPW].Following the terminology of [BGP], a FSCBB, or an Aleksandrov space is acomplete inner metric space with finite Hausdorffdimension, which is curved frombelow in (local) distance comparison sense.Theorem C. Let X be an n-dimensional Aleksandrov space with curv X ≥k,k < ( π4 )2, rad X = 1, and xtn+1 X = xtn+1 Dnk (1). Then(i) There is an isometric embedding of ∆nk in X with totally geodesic interior;(ii) If X is a Poincar´e duality space then k ≤k(n);(iii) If X is a Poincar´e duality space and k = k(n), then X is isometric toD∆nk(n).The essential new technical tool used in the proof of Theorem C is the followinganalogue of the rigidity version of Toponogov’s distance comparison theorem forAleksandrov spaces.Theorem D. Let X be an Aleksandrov space with curv X ≥k.For any pair(p0, c0), where co is a minimal geodesic in X with end points p1, p2 and p0 /∈c0, let(¯p0, ¯c0) be the corresponding pair in S2k, i.e.

dist(pi, pj) = dist(¯pi, ¯pj), 0 ≤i < j ≤2(if k > 0 assume all distances < π/√k). Then for corresponding interior pointsq ∈c0, ¯q ∈¯c0, we have(i) dist(p0, q) ≥d(¯p0, ¯q), and(ii) if equality holds, any minimal geodesic cq from p0 to q spans together withc0 a unique triangular surface isometric to the one spanned by (¯p0, ¯c0) in S2k, andwhose interior is totally geodesic.Part (i) of this global distance comparison theorem was proved in [BGP].

Arigidity statement for hinges as in Riemannian geometry (cf. [CE, G1]) follows byapplying (ii) above twice.In the generality of Theorem C as stated above, we also need a new metriccharacterization of Sn1 .

In order to describe this, let the q-packing radius, packq X,of X be the largest r so that there are q disjoint open r-balls in X, i.e.,2 packq X =max(x1,...,xq) mini

. , xq ∈X.Theorem E. Let X be an n-dimensional Aleksandrov space with curv(X) ≥1.Thenpackn+2(X) ≤packn+2(Sn1 )

4KARSTEN GROVE AND STEEN MARKVORSENand equality holds if and only if X is isometric to Sn1 .This characterization together with Yamaguchi’s fibration theorem [Y] also yieldsthe following pinching theorem of independent interest.Corollary F. For each integer n ≥2 there is an ε = ε(n) > 0 such that anyRiemannian n-manifold M with sec M ≥1 and packn+2(M) ≥packn+2(Sn1 ) −ε isdiffeomorphic to Sn.We conclude with another pinching—or rather recognition theorem. As in ourprevious results above, this is also based on a complete metric classification ofa certain class of Aleksandrov spaces.In this case the class consists of all n-dimensional Aleksandrov spaces X with curv(X) ≥1 and diam(X) = xtn+1(X) =π/2.

Rather than giving the list here, we point out that for each n only two ofthem are Poinc´are duality spaces, namely, RP n1 = Sn1 /Z2 and the double sphericalsimplex D∆n1. From this one derivesTheorem G. For each n ≥2 there is an ε = ε(n) > 0 with the following property.Any closed Riemannian n-manifold M with sec M ≥1 and xtn+1(M) ≥π/2 −ε iseither diffeomorphic to RP n or homeomorphic to Sn.

Moreover, there are metricson M ∼Sn, RP n with sec M ≥1 , diam M ≤π/2, and xtn+1(M) arbitrarily closeto π/2.This result can be viewed as a generalization of the main results in [GP1, OSY].Details and further applications will be published in [GM].It is our pleasure to thank S. Ferry for helpful suggestions related to the proofof Theorem C.References[BGP]Y. Burago, M. Gromov, and G. Perelman, Aleksandrov’s spaces with curvatures boundedfrom below I, Uspekhi Mat. Nauk (to appear).[CE]J.

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International Congressof Mathematicians, Kyoto, Japan 1990, Springer-Verlag, 1991, pp. 511–519.[GM]K.

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98 (1989), 219–228.[PZ]P. Petersen and Zhu, An excess sphere theorem, Ann.

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(to appear).[R]W. Rinow, Die Innere Geometrie der Metrischen R¨aume (1961), Springer-Verlag, 1961.[SY]K.

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(2) 133 (1991), 317–357.Department of Mathematics, University of Maryland, College Park, Maryland20742-0001E-mail address: kng@math.umd.eduMathematics Institute, Technical University of Denmark, 2800 Lyngby, DenmarkE-mail address: steen@mat.dth.dk

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