Math. Z. 218,597~502 (1995)
Mathematische
Zeitschrift
9 Springer-Verlag 1995
On radius, systole, and positive Ricci curvature
Frederick W i l h e l m *
Department of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, NY 11794, USA
(e-mail: wilhelmmath.sunysb.edu)
Received 28 June 1993; in final form 18 October 1993
A closed Riemannian n-manifold M with Ricci M > n - 1 has diameter < 7r
([M]), and equality holds only if M is isometric to the unit sphere S ~ ([Cn]).
Given these results, it is natural to ask whether M is diffeomorphic to S n if the
diameter of M is almost 7r. This question was answered negatively by Anderson
and Otsu, who showed that even the topology of such a space can be different
from the topology of the sphere ([A], [O1]). Thus one must have stronger hypotheses to prove a differentiable sphere theorem. There are already results along
these lines in [O2], [PSZ], [Wm], [Wu], and [Y1]. We will prove generalizations
of all of these theorems and provide a corresponding characterization of real
projective space.
Recall that the radius of a compact metric space, (X, dist) is defined by
Radius X = min max dist(x,
xEX yEX
y).
(Cf [GP2], [SY].)
Theorem 1 (Radius P i n c h i n g Theorem) Let n C N and k c R. There is an
c > 0 so that i f M is a closed Riemannian n-manifold with
Ricci M > n -
1
sec M > k, and
Radius M > 7r - ~,
then M is diffeomorphic to S n. Moreover, the Lipshitz distance between M and
S '~, d i s t L ( M , S n ) , converges to 0 as ~ ~ O.
This generalizes Theorem 3 in [O2], Corollary 2 in [PSZ], and Theorems 9 and
28 in [Wm].
* Supported by a National Science Foundation PostdoctoralFellowship
598
F. Wilhelm
Recall that the first systole of a closed Riemannian manifold, Sys~ M , is the
length of the shortest closed, noncontractible curve. As pointed out in [Wm], a
lower bound for Sysl M implies a lower bound for the radius of the universal
cover of M . This is the main idea in the proof of the following systole pinching
theorem.
Corollary 2 (Systole Pinching Theorem) Let n E N and k E •. There is an
e > 0 so that i f M is a closed Riemannian n-manifold with
Ricci M > > _ n - l ,
secM >k,
and
c~ > Sysl M > Tr - e,
then M is diffeomorphic to RP n. Moreover, l i m ~ 0 diStL(M , RP n) = O.
This generalizes Theorem 10 in [Wm]. Before stating the next Theorem, we
review the various notions of excess.
Definition 3 Given two points p and q in a compact metric space X, the excess
function f o r p and q, ep,q : X ----* R, is given by ep,q(X ) = dist(p, x ) + dist(x, q ) dist (p , q ).
Given p E X and d > O, the d-excess o f X at p is
epa =
max
rain
e,q(x)
xED(p,d)qES(p,d)
e,
w h e r e D ( p , d ) = {x E X s.t. d i s t ( x , p ) <_ d } a n d S ( p , d )
d}. The d-excess o f X is
ed(X) = max e(.
pEX
= {x E X s.t. d i s t ( x , p ) =
t,
The excess o f X is given by
e(X) =
rain
max ep q(X),
(p,q)EXxX xEX
'
and the upper excess o f X is
E (X ) - max min max ep q(X ).
pCX qEX xEX
'
(cf [AG], [GP1], and [02]).
T h e o r e m 4 Let n E N, k E ~, and d > O. There is an e > 0 so that if M is a
closed Riemannian n-manifold with
Ricci M > _ n - 1,
sec M > k,
e d ( M ) < e, and
Diam M >_ 7r - e,
then M is diffeomorphic to S n. Moreover, lime-.odiStL(M, S ~) = O.
On radius, systole, and positive Ricci curvature
599
This generalizes Theorem 4 in [O2], Corollary 7 in [Wm], and a theorem in
[Wu].
A key ingredient in the proofs of Theorems 1 and 4 is the following topological sphere theorem which was recently proven independently by Colding and
Perelman. (See [Co] or [P].)
T h e o r e m 5 Given any n 6 N and k E IR there is an c > 0 so that if M is a closed
Riemannian n - m a n i f o l d with Ricci M > n - 1, sec M >_ k, and D i a m M > 7 r - c,
then M is h o m e o m o r p h i c to S ~.
For the proof of Theorem 1 we first combine Gromov's Precompactness Theorem
(see [Gm 1]) with Yamaguchi's Fibration Theorem (see [Y2]) and conclude that it
is sufficient to prove that any sequence of closed Riemannian n-manifolds which
satisfies
Ricci Mi ~ n -- 1,
sec Mi >_ k,
and
(5)
Radius Mi --+ 7r
converges to S n in the Gromov-Hausdorff topology. By appealing to the proof
of Theorem 3 in [O] (cf also the proof of Theorem 9 in [Wm]) we see that it is
sufficient to show that any limit of such a sequence is n-dimensional. We will
give two proofs of this. This seems justified since both are short, but rely on
previous results whose proofs are quite long. The first proof has the defect that
it fails to work in dimensions 3,7, and 15.
For the first proof, we let Moo be the limit of a sequence which satisfies
(5), and observe that we can use the proof of Lemma 1 in [GPI] to show that
E ( M ~ ) = O.
The Main Theorem and Proposition 22 in [Wm] then imply that Moo is
homeomorphic to a sphere and that for all but finitely many i there is a fibration
fi : Mi
, Moo with connected fiber. By Theorem 5, all but finitely many of the
Mi's a r e homeomorphic to S n. So by the commentary following Theorem 5.1 in
[Br], we see that dim Moo must be n, unless n = 3,7, or 15.
Throughout the remainder of the paper we adopt the convention that if p is
a point in a metric space X and r > 0, then
B ( p , r ) = { x C X s.t. d i s t ( x , p ) < r}.
The idea behind the second proof is to use the argument in [P] to obtain
a uniform lower bound for the filling radii of the Mi's and then to appeal to
Gromov's famous (but difficult to prove) inequality,
FillRad M <_ C ( n ) ( v o l M ) ~ ,
where C ( n ) is a constant which depends only on n. (See [Gm2] for the definition
of filling radius and the proof of this inequality.)
For each point xi 6 Mi let A(xi) C Mi denote the set of points at maximal
distance from xi, and let li C Mi denote the set of points Pi E Mi so that
600
F. Wilhelm
Pi E A(qi) for some qi E Mi. The argument in [P] shows that the distance
function from each point Pi C I i has no critical points in B(pi, Radius Mi), and it
is elementary to show that E(Moo) = 0 implies/co = M~. We therefore obtain a
uniform lower bound for the filling radii of the Mi 'S by appealing to the following
modification of a result of Gromov.
L e m m a 6 Let M be a closed orientable Riemannian n-manifold. Suppose there
is a sequence o f numbers 0 < ~1 < ~2 < . .. < cn+l so that f o r every x C M and
every i <_ n, B (x, Ei + ~2 E'I ) can be contracted inside o f B (x, Ci+l). Then
1
FillRad M >_ =el.
L5
This can be proved by the retraction method in [Gm2], 4.5A, B, and C. (Cf also
[Gm2], 1.2B and C and the proof of Theorem 2 in [K].)
We will omit the proof of Corollary 2 since it is almost identical to the proof
of Theorem 10 in [Wm]. (The only difference is that references to Theorem 1
are substituted for references to other less general radius sphere theorems.)
The proof of Theorem 4 is also via the Gromov-Hausdorff convergence technique. We will show that any sequence of closed, Riemannian n-manifolds, {Mi },
which satisfies
Ricci Mi >_ n - l,
secMi > k
(6)
Diam Mi -+ ~, and
ed(Mi) ~ 0
converges to S" in the Gromov-Hausdorff topology, and then appeal once more to
Gromov's Precompactness Theorem and Yamaguchi's Fibration Theorem. (See
[Gml] and [Y2].)
By Corollary 6 in [Win], any limit Moo of such a sequence is an almost
Riemannian space with positive injectivity radius. Moreover, for all but finitely
many i, there is an almost Riemannian fibration ~ : mi ----* Moo with connected
fiber. (See [Wm] for the definition of almost Riemannian space and almost Riemannian submersion. Cf also Corollary 4.21 in [Y3].)
The next step is to show that M ~ is n-dimensional.'We will provide two
ways of doing this.
For the first, use Lemma 1 in [GP1] to conclude that e ( M ~ ) = 0. Then since
ca(M) is also 0, it follows that Moo is homeomorphic to a sphere. By Theorem
5, all but finitely many of the Mi's are also topological spheres. So by appealing
to [Br] we get dim Moo = n. (Unless n = 3, 7, or 15.)
For the second proof, we first appeal to the argument in [P] to see that if
Pi,qi E Mi satisfy dist(pi,qi) = diameter Mi and i is sufficiently large, then
there are no critical points for the distance function from Pi other than qi. In
particular, B(pi, -~) is contractible. (See [Ch] or [Gr] for a discussion of critical
points of distance functions.)
On radius, systole, and positive Ricci curvature
601
By passing to a subsequence if necessary, we may assume that {Pi} and
{qi} converge to points Poo,qoo C Moo. By Lemma 1 in [GP1], epo~,qrr ~ O. In
particular, B(poo, r) is contractible for all r < 7r.
For i sufficiently large we have f i - l B ( p ~ , 4) C B(pi,-~) C f i - l B ( p ~ , ~-).
This is a contradiction because the inclusion map t : f . - I B ( p ~ , 8 8
>
fi
(P~,
is a homotopy equivalence of spaces with nontrivial, reduced Z2homology which factors through the contractible space B(pi, 2)" So dim Moo = n.
Finally we can argue as in the proof of Theorem 9 in [Wm] (p. 1136) that
vol Moo = v o l S ~ and hence that v o l Mi --~ v o l S ~. The result follows from the
main theorem in [YI].
Remarks: T h e argument in the second proof of Theorem 4 can also be used to
give a third proof of Theorem 1.
All of the above references to the main theorem in [Wm] can be replaced with
references to Remark 4.20 in [Y3].
Acknowledgement. It is my pleasure to thank Grisha Perelman for several enlightening conversations
about this paper. 1 woud especially like to thank him for suggesting the final formulation of Lemma
6, and for pointing out all of the main ideas in the second proof of Theorem 4.
References
[A]
[AG]
[Br]
[Co]
[Ch]
[Cn]
[Gml]
[Gm2]
[Gr]
[GPll
[GP2]
[K]
[M]
[01]
[o21
[P]
M. Anderson, Metrics of positive Ricci curvature with large diameter, Manuscripta Math. 68,
(1990), 405-415.
U. Abresch and D. Gromoll, On complete mani[olds qf nonnegative Ricci curvature, JAMS
3, (1990), 355-374.
W. Browder, Higher torsion in h-spaces, TAMS 108, (1963), 353-375.
T. Colding, Ricci diameter sphere theorem, in preparation.
J. Cheeger, Critical points of distance .[unctions and applications to geometry, Geometric
Topology: Recent developments, Montecatini Term6, 1990 (Ed. P. de Bartolomeis ans F.
Tricerri) Springer Lecture Notes 1504 (1991), 1-38.
S. Y. Cheng, Eigenvalue comparison theorem and geometric applications, Math Z. 143,
(1975), 289-297.
M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. I.H.E.S. 53,
(1981), 53-73.
M. Gromov, Filling Riemannian manifidds, J. Differential Geometry, 18 (1983), 1-147.
K. Grove, Critical point theory for distance functions, Proc. A.M.S. Summer Institute on
Differential Geometry, UCLA, 1990.
K. Grove and P. Petersen, A pinching theoremjor homotopy sphere's, JAMS 3, (1990), 671677.
K. Grove and P. Petersen, Volume comparison h la Alexandrov, Acta. Math., 169, (1992),
131-151.
M. Katz, The Filling radius of two-point homogeneous spaces, J. Differential Geometry, 18,
(1983), 505-511.
S. B. Meyers, Riemannian man!fblds with positive mean curvature, Duke Math. J. 8, (1941),
401-404.
Y. Otsu, On man!fblds of positive Ricci curvature with large diameters, Math. Z. 206, (1991),
255-264.
Y. Otsu, On maniJolds of'small excess, Amer. J. Math., to appear.
G. Perelman, A diameter sphere theorem for manifolds of positive Ricci curvature. Math. Z.,
218, (1995), 595-596.
602
F. Wilhelm
[PSZ] P. Petersen, Z. Shen, and S. Zhu, Manifolds with small excess and bounded curvature, Math.
Z. 212, (1993), 581-585.
[SY] K. Shiohama and T. Yamaguchi, Positively curved manifolds with restricted diameters, Geometry of Manifolds (K. Shiohama, ed.), Prospectives in Math. 8, Academic Press, 1989,
345-350.
[Wm] F. Wilhelm, Collapsing to almost Riemannian spaces, Indiana Univ. Math. J. 41 (1992),
1119-1142.
[Wu] J-Y. Wu, Hausdorffconvergence and sphere theorems, Proc. of Symposia in Pure Mathematics
UCLA 1990.
[Y1] T. Yamaguchi, Lipshitz convergence of mani.[olds t~fpositive Ricci curvature with large volume,
Math. Ann. 284 (1989), 423-436.
[Y2] T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math., 133,
(1991), 317-357.
[Y3] T. Yamaguchi, A convergence theorem in the geometry of Alexandrov space, preprint.
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