Annals of Mathematics
On Complete Open Manifolds of Positive Curvature
Author(s): Detlef Gromoll and Wolfgang Meyer
Source: Annals of Mathematics, Second Series, Vol. 90, No. 1 (Jul., 1969), pp. 75-90
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1970682
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On completeopen manifolds
of positivecurvature
By DETLEF GROMOLL* and WOLFGANG MEYER
Let M be a complete riemannian manifold of dimension n > 2 with sectional curvature Ko > 0 for all tangent planes a. Many effortshave been made
to determine the topological structure of M. The most significant results
were obtained when K0 > a > 0, then M is necessarily compact by a classical
theorem of Myers. In the alternative case, when K is not bounded away
from zero for all planes and hence M open, only very little was known. We
are going to study the latter situation in this paper, and we assume fromnow
on that M is not compact. As a main theoremwe state that M is contractible
and furthermorediffeomorphicwith euclidean space R7 for almost all dimensions. Therefore, the classification problem for complete open manifolds of
positive curvature has a surprisinglyeven solution.
The announced result is related to a result of the firstauthor in [10],
according to which a compact riemannianmanifold of positive curvature with
non-empty convex boundary is diffeomorphicwith the standard disc. While
in that case the boundary condition of convexity has stronger implications
than the compactness of a manifold without boundary, the behavior of the
open manifold M at infinitysomehow replaces that boundary condition. Essentially, the theory of convex sets in riemannian manifolds provides some
new tools and important links.
In the course of our investigation we are also dealing with various other
more geometric aspects such as questions about the isometrygroup of M and
the global behavior of geodesics.
1. Poles and the end structure of M
For all basic concepts and tools in riemannian geometrythat will be used
without comment,we referto [9]. Compare also Preissmann [15] in connection
with this section.
Among the complete open positively curved manifolds, convex hypersurfaces in euclidean space are known best and have been studied intensively in
the theoryof convex bodies. Assume there is an isometricimmersionM-+R ~1.
Then necessarily, M is imbedded as a closed hypersurfaceand bounds a convex
* Miller Fellow
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76
GROMOLL AND MEYER
body diffeomorphicwith the half-space R++1= {a e R7+, anal
O}, M aRn+' =
a
R . This follows fromStoker [18] for n = 2, and Heijenoort [12] in arbitrary
dimensions. In this special situation therefore,M can be realized as the graph
of a non-negative strictlyconcave differentiablefunctionf: RI
R in Rn+1 up
to isometry, where strict concavity of f means in an equivalent formulation,
the hessian form of the second derivatives DDjf, or the second fundamental
form of M with respect to the unit normal vector fieldpointing away from
Rn+1 is positive definiteeverywhere. But, of course, in general an isometric
imbedding of M as hypersurfacein Rn+1 does not exist even locally for n > 3,
and the possibility of realizing M isometricallyin Rn+k with large codimension
k according to the Nash theorem,has not been of any help as yet for intrinsic
problems concerning relations between curvature and topology of riemannian
manifolds. In the case n = 2, isometricimbeddingsof M in R3 do always exist
locally and, as soon as it is known that M is diffeomorphicwith R2, also exist
globally by a theoremof Alexandov in [1]. For details in the foregoingcontext
see Busemann [3].
We call the point p e M a pole if the exponential map expp: Mp M is a
submersion or has maximal rank all over the tangent space Mp of Mat p, then
when M is simply connected.
expp is a covering map and thus a diffeomorphism
We will see that the poles form a compact set P c M. The existence of a
pole in M obviously has very strong topological and geometric consequences.
Unfortunately, P is empty in general, one cannot find a canonical diffeomorphism between the universal riemannian covering space of M and RI by means
of the exponential map or "polar coordinates" as in the Hadamard-Cartan
theorem,where the sectional curvature of M is non-positiveand hence P= M.
Even for surfaces M in R3 which are explicitly definedby simple equations,
it is not at all an easy problem to decide how geodesics behave in the large,
and whether or not some point of M may belong to P. In this respect very
simple manifolds M are furnished by open convex surfaces of second order in
R3, namely paraboloids and hyperboloids. Here the differentialequations for
the geodesics can be integrated in terms of elliptic functions, in special cases
by exponential functions. P is not empty,it always contains the two umbilics
of M, which coincide if M is a surface of revolution. For paraboloids any
other point does not belong to P, but hyperboloidssupply a larger set of poles,
a full neighborhood of the umbilics, with one or two connected components,
depending on the ratio of the axes, compare von Mangoldt [14].
Perhaps the simplest, yet fairly instructive example for a complete open
manifold M of positive curvature, where the following general constructions
can be carried out rather explicitly, is the paraboloid of revolution
-
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MANIFOLDS
(1)
77
OF POSITIVE CURVATURE
inR3
z=x2+y2
with curvature K, = K = (Z + 1)-2, P consists onlyof the umbilic, the origin 0,
as we mentionedbefore. All regular geodesics c: R M can be described in the
followingway. There is exactly one parameter value tosuch that c(to)is closest
to 0. If c(to)= 0, then the image of c is a meridian,the intersectionof M with a
plane in R3containing6(to)and the z-axis. Otherwise c touches the parallel circle
through c(to) and winds around M, intersecting itself infinitelyoften along
either branch of the meridian throughc(to),when to< t oo and to> t
-A,
while the z-coordinate grows very fast monotonically. The restriction of the
geodesic c to [t1,oA) has a conjugate point for any t1< to,but not for t1? to.
We now give an example for a complete open manifold of positive curvature without poles. Consider the paraboloid M definedin (1). By perturbing
the metric of M slightly, we can destroy the injectivity of the exponential
map expo at the only pole 0 in M without creating new poles. Choose some
point q e M and a compact strictly convex metric ball D centered about q,
o 2 D. Deform the metric of M in D a little such that the new manifold M
still has positive curvature and the extension c: [0, A-)+ M of a minimal
geodesic from 0 to q leaves D for ever at the firstpoint c(t1), intersecting the
meridian of M through c(t,) transversally. Now 0 is no longer a pole in M,
since c meets all meridians of M somewhere, neither can a point p # 0 be a
pole in this metric: When p e D, then the infinitecontinuation of a minimal
geodesic fromp to 0 in M is a meridian of M beyond 0, so has a conjugate
point. For p e M - D we can find a branch of a meridian through p which
does not enter D, therefore geodesics starting at p with nearby initial directions will also not enter D, and hence will intersect the meridian.
-
-
From the structure of this example the existence of poles seems to be
extremely rare. However, we shall see later that there are always points in
our situation which replace poles on the loop space level in some weaker
sense. We denote by p(p, q) the metric distance between points p, q e M. A
segment from p to q is a normal minimal geodesic c: [a, ,f] M, c(a) = p,
- a, L(c) arc length of c. A ray or
c(R) = q, 11c(t) 11= 1, p(p, q) = L(c) =
half-line in M with starting point p will be a geodesic c: [a, o) -M, c(a) = p,
such that the restrictionof c to [a, fi] is a segment for any 8 > a, we may
think of c as a shortest connection between p and the point oo at infinityin
the one point compactificationM+ of M. There exists at least one ray from
p to oc. Choose a divergent sequence of points q, e M, in the sense that q,
M. Now the
converges to oo in M+, join p and q, by segments c>: [0, fj]
unit vectors c>(O) have a limit point v in Mp and the geodesic c: [0, oo)
M,
-
-
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GROMOLL AND MEYER
78
oo and p continuous. Or another way
c(t): = exp (tv), is a ray, because f,
R which assigns to each
of saying this, consider the continuous map s: S
point of the unit sphere S c M, the distance fromthe origin to the tangent cut
locus CP of MP in that direction,R two point compactificationof the real numbers. The set s-1( o) of ray directions at p is closed in S and not empty,since
otherwise CP and hence M are compact. Of course, several rays may emanate
from p, even in all directions, if and only if p is a pole. The paraboloid (1)
furnishes an example with exactly one ray fromp to oA,up to parametrization,
p #O.
M that will be a ray when restricted to
A line in M is a geodesic c: R
I a, 00) for arbitrary a e R, in other words, a line is infiniteto both sides and
realizes the distance in M between any two of its image points. In general
lines need not exist, in the case that M simply connected and the sectional
curvature K < O all normal geodesics R EM are lines, the flat cylinderS' x R
has some lines, while its flat non-orientable subcovering, the Moebius strip,
contains no lines at all, neither does the paraboloid (1). And the same fact
carries over to our problem when K > 0. Before going into this we prove a
basic proposition.
M be a normal geodesic and r > 0 along c, r(t)
in
curvature
M
direction
c(t). If r(t0) > 0, then the restriction cr
Ricci
of
of c to [to - z-, t, + z-]has index at least 1 for all sufficientlylarge z > 0.
When furthermore the sectional curvature along c non-negative, K, > 0
with respect to planes a c MCt),c(t) e a, and K, > 0 at t = t0,then the index
of c, is not less than n - 1.
LEMMA 1. Let c: R
PROOF. Consider firstthe linear equation
(2)
I" + aw = 0
R differentiablefunction, a > 0. Call
of Jacobi type on the real line, a: R
its global solution, with the reversed standard initial conditions (p(O) = 1 and
9'(0) = 0, normal. If a(O) > 0, then the normal solution q' of (2) has zeros
- Z1 < 0 < Z2, ( - 1) = q(Z2)
In fact, since a > 0 the smallest positive
= 0.
q'
for example can be estimated fromabove by the zero of the linear
zero of
comparison solution of *" = 0 on [a, 00), where a > 0 is chosen sufficiently
close to 0, (p(a) > 0, p'(a) < 0, and (k(a): = (oa), *'(a): = p'(a). Analogously
one obtains a lower bound for the biggest negative zero of p. There are not
necessarily more than two zeros.
We can assume to = 0. For a: = r/(n - 1) we have the normal solution
p of (2) with zeros - Z1 < 0 < Z2, choose orthonormal parallel vector fields
... , X,1 along c, <Xi, c> = 0. We show, I(pXi, AXE) ? 0 for at least one
Xi *
-
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79
MANIFOLDS OF POSITIVE CURVATURE
1 < i < n - 1, where I denotes the index formof c restricted to [- z,
then the firstassertion is obvious for any r > max (zx, Z2). Since
I~XpXi)
-
I-71
z2j
d
((pt2 -ikp2)d
-k
ki(t): = K, a plane spanned by Xi(t) and c(t), k1++
+ 1ck+
r, we get for
the mean value,
nL- 1
n
)-
-7I1X,
1=
l
11
+
-
-
|71I'
-(
t0 r p) 1dt
n - l 9)td-
d
In order to prove the second statement, choose a differentiable function
m: REAR, m > 0, m(O) > 0, such that K, > m(t) for all a E(t), teR, and
Ko > m(O), a E c(0). The normal solution q' of (2) with a: = m has zeros
- 1 < 0 < Z2. Restrict c again to [- z1, z-2], there the index form I is negative
definiteon the (n - l)-dimensional linear space of vector fieldsqX, where X
is parallel along c, <X, c> = 0. If II XI = 1 and k(t): = Ko, a spanned by X(t)
and c(t), then
I(9X,
X)=
'P
)
-71
<
(9Y2 - kcp2)tdt
(q12
-71
-
mq2)tdt
= 0.
Therefore Ind c > n - 1, whenever r > max (zr, Z-2)
We have as immediate consequence that there exist no lines in M, as soon
as the Ricci curvature is everywhere positive. Furthermore, when Ricci curvature or sectional curvature is non-negative, then r or K must vanish
identically along any line c in M. The same arguments apply to another
situation as well, when M is a Kahler manifoldwith complex structure J and
positive holomorphiccurvature. The vector fieldJ6 is parallel along a geodesic
c: R
M, and the normal solution q' of (2) with respect to the holomorphic
curvature in direction c has two zeros, so M does not contain a line.
We formulatea slightly more general version of Lemma 1, assume K > 0.
In the following Qpqalways denotes the space of all sectionally smooth paths
M fromp to q, as it is used in standard Morse theory.
[0, 1]
-
-
2. Let C c M be compact. There is a compact set D D C such
that any geodesic in f2pq which meets C has index at least n - 1 for all
p, q e M - D.
LEMMA
PROOF. We proceed as in the second part of the proof in Lemma 1, but
choose the functionm universal for all normal geodesics c: R
M, c(0) E C,
this is possible, since C compact. Now we may take for D the set of all points
-
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80
GROMOLL AND MEYER
having distance < r from C.
Associate with any connected non-compact manifold M its set w0(C'0) of
ends in the sense of Freudenthal, r0(oo) can be described as a limit set of the
inverse system wr(M- C) of path connected components for all complements
M - C, C compact. r0(oo) is not empty and may be infinite. We call M connected at infinityif M has only one end, that is to say, for any compact set
C c M there is a compact set D D C such that M - D connected. For example,
euclidean space RI is connected at infinityfor n > 2, a cylinder N x R has
two ends, N compact. Define two sequences to diverge to oo in the same end
of M, if forarbitrarycompact C almost all their points belong to one connected
component of M - C.
With M complete riemannian and p e M, there always exist rays fromp
in any given end, simplytake a limit of segments fromp to p, as before,
to
in that end. Furthermore,two different
where p, is a sequence diverging to
ends A, B e wc(oo)determineat least one line c in M joining A and B. For from
the definitionswe finda compact set C and sequences p,, q, E M - C, divergent
in A, B respectively, so the p, and q, lie in differentpath components of
M - C for all v. Hence there are segments c>: [ - a,, fl] M from p, to
oa, c>(0) e C. The vectors c>(0) have a limit point v in the comql, 0 < a>, f,
M, c(t): = exp (tv), is a
pact set of unit tangent vectors over C, and c: R
wanted line from A to B by an obvious continuity argument for distances
along c. Especially M must contain lines, as soon as M is not connected at
infinity. Combining this fact with Lemma 1, we obtain a result essentially
due to Cohn-Vossen in the case K > 0.
-
-?
-
THEOREM1. A complete open manifold M of positive Ricci curvature
is connected at infinity.
Similarly, a complete non-compactKihler manifoldwith positive holomorphic
curvature has only one end. For example, M= Si, x R does not admit a
complete metric of positive mean curvature, though the sectional curvature
of its natural product metric is non-negative. So, the above result will not
hold for non-negative Ricci curvature in general, even when K > 0 but in
that situation it follows from a theorem of Toponogov in [19]. If M is not
connected at infinity, then M is rigid and splits isometrically into N x R,
where N is compact, thereforeM has at most two ends, compare also [5].
We finallyobserve that the set P of poles in M has to be compact, when
the Ricci curvature is positive or M kihlerian with positive holomorphic
curvature; in fact, whenever the universal riemannian covering M of M does
not contain a line. It sufficesto assume, M simply connected, M = M, since
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81
MANIFOLDS OF POSITIVE CURVATURE
the set of poles in M projects onto P. Thus P = d-'(o), where d: My R
denotes the continuous function that assigns to p e M the injectivity radius
dp of the exponential map expp, and P is a closed subset. We have to show
P is also bounded. Otherwise choose q e M, a divergent sequence of points
q # p, e P, and consider the rays c>: [- a>, oo).-+M, 0< a1. oc, c,(-- a) = pp.
M, c(t): = exp (tv), v e Mq limit point of
c>(0) = q. Now the geodesic c: R
the unit vectors c,(O), furnishes a line in M through q.
-
-
2. Half-spaces and totally convex sets
In [6] and [7], Cohn-Vossen studied in detail the global behavior of geodesics on complete open surfaces of positive curvature, compare also Busemann
[2]. His beautiful and partly delicate analysis makes constant use of the
Gauss-Bonnet formula for polygons. Although such techniques are definitely
confined to surfaces, the most interesting results of Cohn-Vossen do have
generalizations in higher dimensions. The key tool there is a minimumprinciple for distances along geodesics to far points, this will lead us to strong
convexity conditions.
A regular geodesic c: [a, fi] M is called concave with respect to q e M,
if the continuous function s
p(c(s), q) does not assume a weak relative
minimumat an interiorpoint of [a, fi], in this case any restrictionof c is also
concave for q. Our assumption K > 0 now implies that c is necessarily concave
for all points sufficientlyfar away.
LEMMA 3. Let C c M be compact. There exists a compact set D D C such
C in M is concave with respect to all points
that every geodesic c: [a, f]
qeMD.
PROOF. We have only to look again at Lemma 2 from a differentviewpoint. Consider the normal solution q' of (2) with the function a: = m and
choose D for C as in the proof there. Suppose c0: [a, i] C regular geodesic
and s p(c0(s), q) has a weak relative minimumat s, e (a, ,f).
in M, q EM-D,
M, -/> A,c(0) = c0(s0),c(z) = q, K<(0), 0(s0)>= 0,
We finda segment c: [0,a]
by the firstvariation formula. The second variation of arc length for c with
free boundary conditions along c0 and fixed end point q is just given by the
-
-
index formI, since c0geodesic, so I((pX, 9X) < | (q,2 - mqp2))dt= 0 on [0, z2],
X parallel field along c, X(0) = 60(s0). Hence c cannot minimize the distance
fromq to c0locally.
As a firstapplication of this lemma we have, there exists no totally geodesic immersionof a compact manifoldinto M, especially all periodic geodesics
are constant.
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82
GROMOLL AND MEYER
We introduce two basic notions. Call a non-emptysubset A of a complete
riemannian manifold M totally convex, if A contains the image of every
geodesic c E Qpqfor arbitrary p, q e A. Any non-emptyintersection of totally
convex sets is again totally convex. It is not difficultto show that a closed
totally convex set A carries the structure of a k-dimensional topological
submanifold with closed boundary MAin M, 0 < k < n, the interior points
forma totally geodesic submanifold,the possibly empty boundary has unique
supporting (k - l)-dimensional tangent planes at all points of a dense subset
in MA. One can prove, the inclusion A c M is a homotopy equivalence, at
least when A is compact, thereforeM may be the only totally convex subset,
this always happens in the case of compact manifolds, the euclidean sphere
S' furnishes an obvious example. Compare [5] and [11] for furtherdetails.
M be a ray, p: = c(O). Consider the open metric balls
Let c: [0, o)
B,(t): =-{q e MI p(c(t), q) < p(c(t), p)} about c(t) with radius p(c(t), p), t > 0.
It follows immediately fromthe triangle inequality that
-
B,(tl)
( 3)
for 0 < tj <
(--B,(t2)
t2 c
Define the open half-space for c to be the union Be: = U>o B,(t), which may
be viewed as open ball about oo with radius c.
We now return to the case K > 0 and obtain a basic construction for
non-trivial totally convex sets in M.
LEMMA4. The closed complementM
convex.
PROOF. M
-
Be of any half-space Be is totally
Be is not empty, since c(0) X Be. Suppose, there exists a
geodesic c0: [a, ,8] M and c0(a), c0(i3)e M - Be, c0(s0)e Be for some soe (a, ,8).
We find t, > 0, c0(s0)e B,(t,), and t2> t, according to Lemma 3 such that c0
is concave with respect to c(t2). But c0(s0)e Be(t2) by (3), so the function
s
p(c0(s), c(t2)) would assume its absolute minimum at an interiorpoint of
-
].
kxofl
We mention, though Lemma 3 is no longer necessarily true for K > 0,
say with M euclidean space, the last result, which essentially stabilizes the
concavity of far geodesics, remains valid in that situation. One can modify
the above arguments using Toponogov's comparisontheoremforangles, see [5].
Another remark is that M - B. may be compact or not, on the paraboloid (1)
only the pole 0 and all compact metric balls centered about 0 arise as complements of half-spaces, but in general the shape of half-spaces is highly more
complicated, a sheet of the convex rectangular hyperboloid of revolution
provides an example where M - B, unbounded.
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MANIFOLDS OF POSITIVE CURVATURE
83
Now we describe a basic constructionfor compact totally convex sets with
respect to a point p in M. Consider the union B: = UC B, of all half-spaces
(M - B) is a closed
for rays c emanating from p, certainly M - B=
convex set containing p in its boundary. But M - B must also be bounded
and hence compact, because otherwise by convexity we would finda sequence
of segments starting at p and converging to a ray in the closed set M - B,
which is impossible, since all rays fromp to oo run in B. In fact we obtain
a more general engulfingtheorem for compact subsets of M.
fn
LEMMA 5. Any compact set C is contained in a compact totally convex
set D.
There exists a filtration Ci of M, Ci c intCivi U ,,=1Ci = M, Ci compact and
totally convex.
M from
PROOF. For every point q e M - C there is a segment c: [0, ,]
=
=
e
=
L(c)
p
C}
the closed set C to q, c(O) e C, c(R) q, p(C, q) inf {p(p, q) I
=,S.
M a ray from C to cc if the restrictionof c to [0, t]
Call a geodesic c: [0, c)
is a segment from C to c(t) for all t > 0. As before it follows immediately,
such rays do exist. For any sequence c,: [0, 8,B] M of segments from C to
cc, the unit vectors &,(0) have a limit v in the tangent bundle,
since C compact, define c(t): = exp (tv), clearly C c M - B,. Again we form
the union B: = U, B, of open half-spaces with respect to all rays c from C to
(M - B) is closed, totally convex, and C c M - B. If there
cc, M -B
were a sequence q, e M - B, q, I cc, we would find segments from C to q,
and a limit ray to cc in the closed convex set M - B, contradicting the choice
: D compact.
of B, hence M - B
The second part of the assertion is an obvious consequence of the first,
using induction and a filtrationof M by metric balls centered about some
fixed point p.
There are even continuous filtrations of M by compact totally convex
sets, see [4], [5]. Only a very few totally convex sets can be obtained from
the canonical construction in Lemma 5, the totally convex hull of any point
p # 0 on the paraboloid (1) for example, never arises.
Some totally convex sets are of specific significance. Suppose, p e M is
totally convex. We will call p a simple point in M, because the exponential map
expp covers p just once. The existence of such a point has strong topological
implications, the loop space Qppand hence M must be contractible by standard
Morse theory. We are now in position to get complete informationabout the
homotopy type of M.
-
fn,
THEOREM2. The set S of simple points in M intersects every compact
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84
GROMOLL AND MEYER
totally convex set C. Therefore S # 0 and M is contractible.
PROOF. Choose a compact set D D C according to Lemma 3. Since C is
compact, we can definea mapping h: M-DISC, p(q, h(q)): sup{p(q,p) p e C}.
Clearly h(q) is unique and simple because of the concavity of all geodesics in
C with respect to q.
We shall later discuss furtherquestions about the topological structure
of M, but append some other remarks in this context. The map h in the last
proof, which associates with every point sufficientlyfar away from C its
farthest point in C, is non-constant in general and continuous. There exists
a distinguished simple point in C at maximal distance from the boundary AC,
the soul of C in the terminologyof [4], [5]. Notice that S contains the compact
set P of poles as a proper subset by Lemma 7. So there are always points p
in M whose tangent cut locus is not emptyand intersects the conjugate locus
at a point closest to the origin of Mp. Compare results and questions of
Weinstein [20] for compact manifolds in this connection.
3. Symmetries and the global geometry of geodesics
This section deals with some geometric problems that are of particular
interest. Consider the group G of isometries with the compact-open topology,
acting on M. G is a not necessarily connected Lie group by the general
theorem of Myers-Steenrod. Contrary to the case K i 0, it will turn out that
G has to be compact. Note first,G cannot act transitively on M, since K is
not bounded away fromzero. Though M is never homogeneous, large isometry
groups may occur. No discrete subgroup F c G acts freelyon M, otherwise the
orbit space M/F with the induced riemannian structure would be a non-compact complete manifold of positive curvature again, hence M/F is contractible
according to Theorem 2, but the fundamental group of M/F is just F. Moreover, we prove a stronger result.
THEOREM3. There exists a filtration C, of M as in Lemma 5 such that
Ci compact, totally convex, and G-invariant. G is compact and has a fixed
point in every closed invariant totally convex subset of M.
PROOF. For the firstpart of the theorem it sufficesto show, any compact
set C is contained in a compact totally convex set invariant under G. Consider
the closed orbit A: = G C and the union B: = U, B, of open half-spaces with
respect to all rays c fromA to oo. Certainly M - B =fn (M - B,) is closed,
totally convex, G-invariant, and A c M - B. But M - B is also compact.
First of all M - B c G *DP for almost all integers v > 0, D, denotes the compact
metric ball of radius v centered about some point in C, D, D C. Otherwise we
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85
MANIFOLDS OF POSITIVE CURVATURE
could select a sequence q, eM - B - G*Di, p(A, q,) a Oo, and points p, e A,
p(p?, q,) = p(A, qj), and g, e G such that g,(p,) e C. So we would have segments
M - B fromA to g,(q,), c,(O) =gp(pp),cp(fSl) =g,(q,), and a limit
c,: [O,8fl
ray c fromA to A, in contradiction to the choice of B. The curvature K is
bounded away from zero on G D,, thereforethe diameter of M - B must be
finite,because any two points can be joined by a segment in M - B.
Since compact G-invariant sets do exist, G is compact. It remains to
construct a fixed point of G in any closed totally convex set A c M, we may
assume that A compact. G leaves the boundary AAof the topological manifold
A invariant, so the unique point p farthest away fromAA in A, p(p, AA)
sup {p(q, AA) I q e A}, is fixed under G and also simple.
There is a parallel classical theorem of E. Cartan about fixed points for
compact groups of isometries acting on a complete simplyconnected manifold
of negative sectional curvature, see [13]. For any fixed point p of G we get a
faithful representation G
0(n) by mapping g e G into its differentialat p,
0(n) is viewed as the orthogonal group of MP. In the most symmetric case,
0
G
0(n), one can easily construct an 0(n)-invariant isometric imbedding of
M as convex hypersurface of revolution in euclidean space Rn+'.
Now we turn to the study of geodesics and start with deriving a fundamental estimate.
-
-
LEMMA 6. Given any compact set C c M, there exists a bound
that the length of any geodesic in C is less then X.
X
such
PROOF. Suppose we finda sequence of normal geodesics c,: [-v, y]
C.
The vectors j,(O) have a limit point v in the compact set of unit tangent
vectors over C, and c: R E M, c(t): = exp (tv), is a geodesic running in C.
Consider the compact closure A of c(R) in M, A c C. We choose a set D D C
according to Lemma 3, q e M - D, and p e A, p(p, q) = p(A, q). There is a
sequence t, e R such that c(t,)
p, let w e MP be a limit point of the vectors
The
M,
R
geodesic c:
c(s): =exp (sw), runs in A passing through p,
c(t,1).
but this contradicts the local concavity of c at p for q.
-
-
M is proper,
THEOREM4. The exponential map exp,: MI
4)
limvOO.
exp, (v) =
.o
PROOF. Take a filtrationCi of M as in Lemma 5, p e C1, and determine
uniform bounds xi for the lengths of geodesics in Ci according to Lemma 6.
Let v, e MP be a sequence, vpA, and consider the geodesics c,: [0, 1]
My
=
an
find
we
Given
such
v
integer
i,
any
cso.
vi
c,(t):-exp (tv,), L(c,)
II
+
that IIvp II > Xi for all v > vi. Since C, totally convex, exp (vp)e Ci implies
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86
GROMOLL AND MEYER
that c, runs in Ct, so
vs II < xi. Hence exp(v,) e M - Ct, v > of, and
limps exp, (vp) = co.
In particular, (4) tells us that both branches of any non-constantgeodesic
c: R oM go to infinity,
( 5)
lim, ?+. c(t)
=c, limoo
c(t) =
co
This solves a problem of Chern posed in [8].
As another corollary of Theorem 4 we have the number of geodesics in
q2pqis finitefor all p, q e M, except possibly when 12pqdegenerate or q critical
value of expp. It is well known fromSard's theorem that for given p the set
of points q where Qpqdegenerates, has measure zero in M.
The last result implies again that M contractible, see Serre [16]. Of
course, in general there may be infinitelymany geodesics in a non-degenerate
loop space 12pq of a contractible complete riemannian manifold, already when
K > 0. The example of the paraboloid (1) shows that the bound for the
number of geodesics in Qpqneed not be uniform in p, q. It follows from the
Morse inequalities that = (-)"c,
= 1, where c, the number of geodesics
of index v in Qpq, hence the total number of geodesics in f~pqis always odd.
Combining this with Lemma 2 and (5), we find a compact set D c M such
that for arbitrary p e M - D there is an unbounded subset of points q e M
for which &lpqcontains at least three geodesics.
We do not emphasize a more quantitative approach to the global behavior
of geodesics in this paper, but a few furtherremarks may be interesting. If
there exists a pole p in M, then it follows easily fromthe proof of Lemma 2
that the differentiablefunction t p(p, c(t)) has positive second derivative,
hence is strictly concave on R, compare also [15]. In general, when the set
P of poles empty, a similar statement need no longer be true. However, by
looking at geodesics fromthe inverse viewpoint at infinityand using the concavity of far geodesics, one can construct a continuous functionA: M-M[O, oc),
*(p)=- 0 for some simple point p e S, and * is concave in the sense that
- * o c strictly convex on R for any non-constantgeodesic c: R
M. The set
Cs: = *-'[0, s] is compact totally convex and p(q, aCS2) = i(S2) (s1) for all
q e aCsi, 0 < SI <? 2. Moreover, the first derivatives of * and the laplacian
A* exist almost everywhere, A-v > 0. For furtherdetails see [5].
-
-
LEMMA7. The set S of simple points is open in M.
PROOF. M - S is closed. Take any sequence p, e M - Sy pp e M, and
choose C compact totally convex, p, e C for all v. Now we have geodesic
loops Ce&
E fp~p, V,: 6z&(O)# 0, and II vI bounded by Lemma 6. On the other
hand, the sequence v II is obviously bounded away fromzero, since almost all
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MANIFOLDS OF POSITIVE CURVATURE
87
p, are contained in a strictly convex neighborhoodof p. Hence there exists a
limit point v ? 0 of v, in MP, and c: [0, 1]-M, c(t): = exp (tv), is a non-trivial
so p e M - S.
geodesic loop in i)pp,
We do not know very much about S. The fact that S open was stated by
Cohn-Vossen [7] in the case of surfaces, but he never published his presumably
more complicated proof. One obtains immediately, from the Gauss-Bonnet
theorem, S = M if the total curvature of M is not larger than w, otherwise
Cohn-Vossen showed that S has to be bounded. Further investigations in
higher dimensionswould be interesting. There is another set of points which
seems to be of some importance for the geometry of M. We call a ray
c: [a, o) ) M extremal and c(a) an extremal point if c cannot be extended to
[a., oo) as a ray for any a0 < a. Since M does not contain lines, every ray
determines an extremal point. There may exist only one such point p in M
as on the paraboloid (1), in this very rare case p is a pole and P = {p}. In a
vague sense, the set of extremal points can be viewed as the cut locus from
infinity,the extremal point c(a) has similar propertiesas a finitecut point. If
c(a) and oo are not conjugate with respect to c, that is to say, the continuation
of c to [a., Ao) has no conjugate points for some a0 < a, then another ray
starts at c(a) in a differentdirection. Moreover, using arguments fairly
analogous to those of Berger's well-knownlemma, compare [9, Lemma 7.8.1],
one can show that there exists a simple point p e S such that for all directions
v e MP we finda ray c: [0, cc) -*M with <c(0), v> > 0. Hence, either there is
a double ray based at p (two rays fromp to oo making the angle w) or at least
three distinct rays emanate from p. One may choose for p a point farthest
away from 0o, namely the soul of any compact totally convex set C arising
fromthe basic construction with respect to a point.
4. The differentiable structure of M
We have seen in ? 2 that M is a contractible manifold. Without further
information,this does not necessarily imply M has the same topological type
as euclidean space R". There exist even uncountably many contractible 4dimensional manifolds which are mutually not homeomorphic,they all differ
in their behavior at infinity. However, in our situation we will prove
THEOREM 5. M is diffeomorphicto Ra.
For two dimensions this result is due to Cohn-Vossen, and follows from
the contractibilityof M via the classification of surfaces. In general M is
called simply connected at infinityif for any compact set C there exists a
compact set D D C in M such that M - D is simply connected. Similarly to
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GROMOLL AND MEYER
88
the set r0(ao) of ends one can always define the fundamental group r1(Cc) of
M at infinitywhen M has one end, but we do not need the entire concept.
Euclidean space RI is simply connected at infinityif n > 3, but not for n = 2.
Now by the affirmativeanswer to the generalized Poincare conjecture, a
contractible differentiablemanifold M is diffeomorphicto RI if M simply
connected at infinity(at least for n > 5), compare Stallings [17]. By applying
this theorem to our problem, we shift some of the difficultiesto topology,
which is not really necessary. In fact, Theorem 5 can be proved directly for
all dimensionsusing special continuous filtrationsof M by totally convex sets
and explicit geometricconstructions;in particular, it turns out that all compact
totally convex sets are topological discs. This approach is somehow more
satisfactory, but technically too involved for our purposes here; it will be
presented within the frameworkof subsequent papers, see [4], [5], [11].
In order to prove Theorem 5 (for n > 5), we show M is simply connected
at infinity,which according to Lemma 5 follows from
LEMMA 8. The complement M - C of any compact totally convex subset
C of M is k-connectedfor 0 < k < n - 2.
M - C be a continuous map representinga homotopy
class of Wk(M - C). Determine s > 0 such that all open metric balls of radius
s centered about points in the neighborhood V ={q I p(C, q) < s} of C are
0. Givenq e V, we find
strictlyconvex in the sense of [9] and f(Sk) n v
a unique point h(q) e C, p(C, q) = p(h(q), q), for convexity reasons. Clearly
A: V
C is a continuous retraction. Now we have a continuous vector field
Y on V - C, Yq: = (p(C, q) - s)(exp I Uq)-'o h(q) # 0, where Uq the open ball
of radius s centered about the origin in the tangent space Mq. By a standard
argument we find a compact smooth n-dimensional submanifold W c V,
C c int W, and fixingsome point p e C, we may extend Y I a W continuously
to a vector field Zon W, Z IW = Y IW, Zq # 0 for q # p. Define a vector
0.
fieldXonM, X W: = Z,XI V- W: = YI V- W,X M- V:
There exists a continuation F: D k+l M of f, Fl Sk = f, p 2 F(Dk+1), for M
contractible and k + 1 < n. Consider Ft: Dkk+l ) M, Ft(a): = exp (t Xo F(a)),
Ft I Sk = f, t > 0. Now Lemma 5 and Lemma 6 yield Ft(Dk+l) c M - C for
large t, notice that II X is bounded away from zero on F(Dk+l) n W and
exp (tXq) i C forall q e M - W, t > 0, since C is totally convex. This completes
the proof.
Our study of complete positively curved manifolds has been a qualitative
analysis in a large part. Many furthernatural questions arise concerning the
global metric structure of such riemannian spaces, we only mention one of
PROOF. Let f: Sk
-
-
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MANIFOLDS
OF POSITIVE CURVATURE
89
them. The curvature of M seems to be concentrated in finitedomains. What
is a proper and more precise description of this phenomenon? For example,
the total curvature of M always exists and does not exceed the area 2wrof the
standard euclidean hemispherein the case of surfaces. It might be reasonable
to conjecture that an analogous result holds in even dimensions for the total
curvature with respect to the Chern integrand.
The basic constructions of this paper have been further developed and
generalized by Cheeger and one of the authors. In connection with some
other new techniques, they also provide a key to the more complicated structure theory for complete manifolds of non-negative curvature, see [4], [5].
Whether similar or modifiedmethods may be applied to manifolds of positive
Ricci curvature as well is as yet rather doubtful since positive mean curvature
only leads to much weaker partial convexity conditions,as does positive holomorphic curvature for Kahler manifolds.
UNIVERSITY
OF CALIFORNIA, BERKELEY.
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