A Radius Sphere Theorem
by
Karsten Grove1 and Peter Petersen2
Introduction
The purpose of this paper is to present an optimal sphere theorem for metric spaces
analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter
Sphere Theorem in riemannian geometry.
There has lately been considerable interest in studying spaces which are more singular
than riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff
limits of riemannian manifolds are almost never riemannian manifolds, but usually only
inner metric spaces with various nice properties. The kind of spaces we wish to study here
are the so-called Alexandrov spaces. Alexandrov spaces are finite dimensional inner metric
spaces with a lower curvature bound in the distance comparison sense. This definition
might seem a little ambiguous since there are many ways in which one can define finite
dimensionality and lower curvature bounds. The foundational work by Plaut in [PI],
however, shows that these different possibilities for definitions are equivalent. The
structure of Alexandrov spaces was studied in [BGP], [PI] and [P]. In particular if X is an
Alexandrov space and p e X then the space of directions E p at p is an Alexandrov space
of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is
homeomorphic to the linear cone over £„ . One of the important implications of this is that
the local structure of n-dimensional Alexandrov spaces is determined by the structure of
(n-l)-dimensional Alexandrov spaces with curvature > 1. Sphere theorems in this context
seem to be particularly interesting. For if one can give geometric characterizations of
Supported in part by the NSF
2
Supported in part by the NSF, the Alfred P. Sloan Foundation and an NSF Young Investigator Award
1
spheres, then one can also characterize manifold points in Alexandrov spaces.
In [P] there is a generalization of the Diameter Sphere Theorem (see [GS]): If X is an
Alexandrov space with curvature > 1 and diameter > -z then X is a suspension. Closed
hemispheres or suspensions over projective spaces, however, are examples of suspensions
that have curvature > 1 and diameter rc, but which are not spheres. So to get a sphere
theorem for Alexandrov spaces we therefore need a stronger invariant than the diameter.
The radius of a metric space X is defined as: radX = min max dip , q) , where d( , )
pe
XqeX
denotes the distance function on X (see [GP1], [GM] and [SY] for other results using the
radius concept). If therefore radX > r then for every point p e X there is q e X such that
d(p,q) > r. With this behind us we can now state our main result as.
Main
Theorem: Let X
be an ^-dimensional Alexandrov space with
curvature > 1 and radius > - then X is homeomorphic to the n-sphere Sn .
This theorem is optimal in the sense that the radius condition cannot be relaxed to a
condition on diameter or to the condition that radius > T . To see this just note that the
above mentioned examples have radius = -z (see also section 2 for more examples). In
[GW] two different optimal sphere theorems for ^-dimensional Alexandrov spaces with
curvature > 1 are proved. In the first one it is assumed that Packn + 2(X) > j . This
condition is easily seen to imply our radius condition, so in this case our result is more
general. In the second result it is assumed that X has no boundary and that Packn(X) > j .
These conditions are, however, neither stronger nor weaker than our assumption.
The rest of the paper is divided into two sections. In the first section we prove the
Main Theorem. In the second section we give some general procedures for constructing
Alexandrov spaces with curvature > 1 and big radius. The abundance of examples we
construct will show that it is hard to get a grip on those spaces which have diameter or
radius equal to -z . This is somewhat in contrast with the manifold situation, where all
2
riemannian manifolds with sectional curvature > 1 and diameter = — have been classified
(see [GG1]).
A c k n o w l e d g e m e n t s : The authors would like to thank M. Bestvina, R.
Edwards, and S. Ferry for helpful conversations concerning the topological aspects of this
paper. Their advise has greatly simplified the topological aspects of our proof.
1 The Antipodal Map
For the rest of this section we will assume that X is an n-dimensional Alexandrov
space with curvature > 1 and radius > -r . If we fix a point p e X then our radius condition
implies that the complement to the open - -ball around p, Cip) = X-Bip , % 12) is a closed
set with non empty interior. Toponogov's Distance (or Angle) Comparison Theorem (see
n
[TJ, [PI], [BGP]) then shows that C(p) is convex and that the distance function dip, •) is
convex on Cip) and strictly convex on the interior of Cip). Thus there is a unique point
Aip) at maximal distance from p. Note that the map p -» Aip) is clearly continuous. We
call this map the antipodal map for X, since on the standard sphere A is the antipodal map.
As observed in [GW] it follows from [P] that X is a sphere if we can show that there is a p
such that the space of directions at A (p) is a sphere or equivalently that A (p) is a manifold
point We prove this by contradiction. Denote by S c X the set of non manifold points in X,
thus we assume that A: X —» S. It follows from the topological stratification of X described
in [P] that S is a closed subset of dimension < n - 1 . Furthermore the dimension can only
be n-1 if X has boundary. Now any Alexandrov space with curvature > 1 and non empty
boundary must have radius < j (see eg [P]). Thus our radius condition implies that dimS <
n-2. We will show in lemma 1 and 2 below that the Alexander-Spanier cohomology
group Hn(X, Z 2 ) = Z 2 and that A = A-A is homotopic to the identity map id:X^
X. Thus
"")
(
A2 induces a map on Hn(X ,Z2), which on one hand is the identity map, and on the other
hand is zero since A2 factors through S which has dimension < n. Hence we get a
contradiction.
L e m m a 1: LetX be a compact Alexandrov space without boundary. Then X has a
fundamental class in Alexander-Spanier cohomology with Z 2 coefficients ie. Hn(X, Z 2 ) =
Z2 .
Proof: We use Alexander-Spanier cohomology as it is described in [S]. Denote again
by S the set of non manifold points. Since X doesn't have any boundary it follows from
the topological stratification results in [P] that dimS < n-2. Then X-S is a connected ndimensional manifold. Therefore the top cohomology class with compact support satisfies
H"(X-S , Z 2 ) = Z 2 . Now the advantage of using Alexander-Spanier cohomology is that
Hn(X , S , Z 2 ) = H"(X-S , Z 2 ) as long as X and S are compact. Using the long exact
sequence for the pair (X, S) now yields:
0 = Hn'\S , Z2) -> Hn(X ,S,Z2)->
Hn(X , Z2) -> H\S , Z2) = 0
Because S has dimension < n-2. Thus we get the desired conclusion. D
L e m m a 2: LetX be as in the Main Theorem and A the antipodal map described
above. Then A2 is homotopic to the identity map on X.
Proof: All of the results on ANR's and decompositions we are going to need for the
proof can be found in [D].
The sets C(p) = X-B(p, n/2) are compact and vary continuously in the Hausdorff
metric. Furthermore it is proved in [P] that they are Alexandrov spaces which are
contractible. Now any Alexandrov space is an ANR since it is finite dimensional and
locally contractible. Thus the sets C(p) are cell-like. Consider now the decomposition G on
j
XxX consisting of the sets {(x, y)}, where y (. C(A(x)), and {(x , C(A(x))}. This is an
upper semicontinuous cell-like decomposition, since the sets C(A(x)) are cell-like and vary
continuously with x. Let/?: XxX -» XYXJG be the natural projection and pz: XxX —»X the
projection onto the second factor. Define the maps i:X —> XxX as i(x) = (x,x) and / : X —>
XxX as f(x) = (x , A2(x)). From the construction of A we see that x, A2(x) € C(A(x)).
Thus the two maps pf and pi are identical. But then also / and / are homotopic since their
domain is a finite dimensional ANR and p: XxX -> XxXIG is a cell-like map, and hence a
(fine) homotopy equivalence. Thus also A = p^f and id- p^i must be homotopic which
is what we wanted to prove.
•
Now that we know X is a sphere we can use this to get more information about the
geometry of X J u s t as was done in [GP1] for riemannian manifolds with sectional
curvature > 1 and radius > - . The important results are summerized in:
Theorem
3:LetX be an rc-dimensional Alexandrov space with curvature > 1 and
radius > -= , then the following statements are true:
1) The "antipodal map" A : X —» X is onto.
2) Any ball B(p , r) where r < radX, is contractible.
3)The n-dimensional Hausdorff measure volX > cn > 0, where c^ is a constant
depending only on n.
4) There is an En > 0, such that if in addition radX > n — £„ then Xis bi-Lipschitz
equivalent to the standard sphere Sn.
Proof: 1) Suppose we have identified X with the standard sphere by some
homeomorphism. Since the map A doesn't have any fixed points it must be homotopic to
the standard antipodal map. In particular A has nonzero degree and must therefore be onto.
2) If p € X then we can find x e X such that p = A (x). Hence the distance function
5
from p has only p and A(p) as critical points. This implies that all balls B(p , r), where r <
radX" < dip , A(p)) are contractible (see [P]).
3) Now that there is a bound for the contractibility radius for X it is easy to get a lower
bound for the filling radius for X (in fact fillradX > radX/(2rc+4)) (see [G]). Gromov has
informed us that the important inequality (fillradX)" < Cnvo]X is true for the spaces under
consideration here. Thus we get a lower volume bound for X.
4) This is proved by contradiction. Suppose that we have a sequence Xt where radX,-» it. By passing to a subsequence if necessary we can suppose that X,- converges to a
space X with curvature > 1 and radius = n. This implies that X is isometric to a sphere (see
eg [GP2]). The fact that all of the spaces X,- have a uniform lower volume bound implies
that the limit X has dimension n = dimX,-. So the sequence converges to the standard
sphere of the same dimension. Then the results in [BGP] (cf also [SOY]) show that the
X,'s must eventually be bi-Lipschitz equivalent to S".
2 Equality Discussion
The Diameter-Suspension Theorem of [P] can be phrased as follows: An ndimensional Alexandrov space X admits an Alexandrov structure with curvX > 1 and
diamX> — if and only if Xis the suspension of an (n-l)-dimensional Alexandrov space E
with curvls > 1. Together with the main result of this paper, this gives a complete
description of Alexandrov spaces with curvature > 1 and diameter > — or radius > -z .
Since both results are optimal it is natural to investigate exactly how they fail, when the
strict inequality is replaced by equality. This is already quite difficult in the case of
riemannian manifolds (cf [GGl ,2]) and the strategy used there seemes to work only for the
first few steps. Specifically, ifXis an Alexandrov space with curvX > 1 and diameter = —
one finds two convex subsets A, B c X at maximal distance, and X is homeomorphic to the
6
union of e-neighborhoods D^A, DEB glued together along their common boundaries
~ dDeB . This fact uses Perelman's extension of critical point theory to Alexandrov spaces.
Although the Rigidity Comparison Theorem of [GM] replaces the use of parallel transport
in the riemannian setting, it does not have the same impact There are several reasons for
this, one of the more fundamental ones being that the space of directions in a riemannian
manifold is the unit sphere, whereas it can be any positivily curved Alexandrov space in
general.
In this connection, it is interesting to point out that as a consequence of our Main
Theorem, any space of directions of an Alexandrov space, X with curvX > 1 and radZ > —,
is topologically a sphere. If this could be proved directely, an easy induction argument
would yield our result as well. We have been informed, by Perelman, that this has indeed
recently been done by Petrunin. The proof is based on a very natural, yet difficult and
involved, notion of (pre)quasigeodesics. With this in mind one might hope that X is a
manifold if radX = —. This, however, is not correct as we shall see below.
Except for the Cayley plane CaP all known examples of Alexandrov spaces Z with
curvZ > 1 and diamZ = y or radZ = ~ are based on the topological join construction. As
noted in [GM] the join Z = X*Y of Alexandrov spaces X and Y with curvature > 1 is again
an Alexandrov space with curvature > 1. Here we give an explicit description of this socalled spherical join:
If X is an Alexandrov space with curvature >1 then the linear cone L(X) is an
Alexandrov space with curvature > 0, such that the space of directions at the vertex is X. If
we have two Alexandrov spaces X, Y with curvature > 1 then we get an Alexandrov space
L(X)xL(Y). The space of directions at the point which is (vertex, vertex) is an Alexandrov
space with curvature > 1 which can be naturally identified with the join X*Y. Note that X, Y
are isometrically imbedded in X* Y in such a way that all points in X have distance j to all
points in Y. Moreover if A c X and B c Y are Alexandrov spaces isometrically embedded
in Xand Y, then A*B c X*Y isometrically. This can be used to compute distances between
7
points in joins in the following way: Any point in X*Y-(X<uY) has a unique coordinate
representation as (x, t, y) where x e X, y e F, t € ( 0 , 7t/2), so if we have two points in the
join let A be a segment joning the x-coordinates and B a segment joining the y-coordinates,
the distance between the original points can now be computed in A*B c S (1) =
5'1(1)*51(1), because A and B can be regarded as isometrically embedded in Sl since they
have length < it. When F is a point we get the spherical cone over X, and when F is the
two point set we get the spherical suspension over X. In all other cases F is assumed to be
connected and have dimension > 1. With this description of the join it is easy to see that:
I 71
(i) diamX*F= max-j -r , diamX, di
It is also possible to compute radX*F, however, here we only point out the following
facts:
(ii) radX*F> § «»min{radX, radF} > |
(iii) radX*F< | « • max{radX, radF} < |
Note in particular that the classes of spaces with curvature > 1 and diameter = —
and/or radius = - is closed under the join operation.
Another important feature of the join operation is that if a compact group G acts by
isometries on both X and Fthen G also acts by isometries on X*Y. Thus we can form the
quotient space (X*Y)IG which generically is different from (X/G)*(Y/G)
This is
interesting because:
(iv) diam(X*Y)IG > \
(v) rad(X* Y)IG > rad(X/G)*(F/G)
Here (iv) is trivial, whereas (v) follows from the Rigidity Distance Comparison
Theorem of [GM]. The above constructions yield numerous examples of spaces with
curvature > 1 and diameter = - or even radius = -r. For the latter, let us point out the
following simple but interesting explicit example:
Let M1 be one of the Wallach examples M = SU(3)/Sl (see [AW]). Then M admits an
Sx action with M/Sl = SU{3)lf1 = N6 the flag manifold. From the above inequalities we
g
7'
derive that radX = f for X = (5 2 " +1 *M 7 )/5 1 . Note that here the convex sets A = €Pn
and
B = N6 are manifolds but X is not.
Similarly radF = § , where Y= {{S2n+l*M1)*{S'ln+l*M1))ISx
. Here the convex sets
are both the space X from above.
These examples illustrate some of the difficulties we discussed above in applying the
methods of [GG1,2] to Alexandrov spaces.
References
[AW]
S. Aloff & N. L. Wallach, An infinite family of 7-manifolds admitting
positively curved Riemannian structures, BAMS 81 (1975), 93-97.
[BGP]
Yu. Burago, M. Grornov & G. Perelman, A. D. Alexandrov's spaces with
curvatures bounded from below, to appear in Uspekhi Mat. Nauk.
[D]
R. J. Daverman, Decompositions of manifolds, Academic Press, New
York, 1986.
[GG1]
D. Gromoll & K. Grove, A generalization ofBerger's rigidity theorem for
positively curved manifolds, Ann. Sci. Ecole Norm. Sup. 20 (1987), 227239.
[GG2]
D. Gromoll & K. Grove, The low-dimensional metric foliations of
euclidean spheres, J. Diff. Geo. 28 (1988), 143-156.
[G]
M. Gromov, Filling Riemannian manifolds, J. Diff. Geo. 18 (1983), 1-148.
[GM]
K. Grove & S. Markvorsen, Metric invariants for the riemannian
recognition program via Alexandrov geometry, preprint.
[GP1]
K. Grove & P. Petersen, Volume comparison a la Aleksandrov, to appear in
Acta. Math.
9
[GP2]
K. Grove & P. Petersen, On the excess of metric spaces and
manifolds, preprint.
[GS]
K. Grove & K. Shiohama, A generalized sphere theorem, Ann. Math. 106
(1977), 201-211.
[GW]
K. Grove & F. Wilhelm, Hard and soft packing radius$g&sS£ theorems, in
preparation.
[SOY]
Y. Otsu, K. Shiohama & T. Yamaguchi, A new version of differentiable
sphere theorem, InvL Math. 98 (1989), 219-228.
[P]
G. Perelman, Alexandrov's spaces with curvatures bounded from below II,
preprint.
[PI]
C. Plaut, Spaces ofWald curvature bounded below, announcement.
[S Y]
K. Shiohama & T. Yamaguchi, Positively curved manifolds with restricted
diameters, in Geometry of manifolds (ed. K. Shiohama). Perspectives in
Math. 8 Academic Press, 1989, pp. 35-350.
[S]
E. H. Spanier, Algebraic Topology, Springer Verlag, 1966.
[T]
V. Toponogov, Riemannian spaces with curvatures bounded below,
Uspekhi Mat. Nauk 14 (1959).
Karsten Grove
Dept. of Mathematics
Univ. of Maryland
College Park, MD 20742
Peter Petersen
Dept. of Mathematics
Univ. of California
Los Angeles, CA 90024
10