Metric deformations of curvature

P A U L E H R L I C H

M E T R I C D E F O R M A T I O N S O F C U R V A T U R E

I: Local Convex Deformations*

1. I N T R O D U C T I O N

In [1], Aubin stated a theorem which implied as a corollary that if a manifold M admits a Riemannian metric with nonnegative Ricci curvature and all Ricci curvatures positive at some point, then M admits a metric of everywhere positive Ricci curvature. It appears the proof in [1] is incomplete and the uniformity and correctness of Aubin's estimates even in the compact case are not clear. While Aubin's estimates somehow rely explicitly on a bound for the sectional curvature, the estimates for our method of local convex deformations do not explicitly involve the sectional curvature. In the non-compact case, Aubin did not consider whether a complete non- negatively Ricci curved metric would remain complete after perturbing the metric to be positively Ricci curved. In view of the well known result of Gromov that any non-compact manifold admits an (incomplete) metric of positive sectional curvature, a Ricci curvature deformation theorem in the noncompact case is of interest only if completeness is preserved.

Motivated by what we thought was the general method suggested by Aubin's paper, we decided to study the following problem: can we find a standard deformation g(t) defined on a metric disk D agreeing with a given metric in M - D to spread positive Ricci and/or sectional curvature from near the center of D to all of D? Motivated by [5] and a conversation with James Simons, we considered a weaker question: can we find a deformation g(t) of a given metric go with positive first derivative of Ricci or sectional curvature near Bd(D)?

This question led to the study of 'local convex deformations' which we discuss in Section 2 along with some technical lemmas used later.

In Section 3 we make some observations on the space of Riemannian metrics. First we need to make precise the idea that for close metrics, metric balls centered at the same point are similar. The precise statements here will be used in proving the Ricci curvature deformation theorems in Section 5. Second we remark that the convexity radius and the injectivity radius functions are locally minorized on suitable subsets of the space of Riemannian metrics.

In Section 4 we study the first derivative Ric' of the Ricci curvature for

* Supported by SFB 40 at Bonn University while paper revised.

Geometriae Dedicata 5 (1976) 1-23. All Rights Reserved Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland

2 P A U L E H R L I C H

all local convex deformations through C 4 metrics with support in D. Since such a local convex deformation may be written in D as g(t)=go +toah where o=distance to Bd(D) is essentially the smoothing function which forces g(t) =go in M - D, all that is needed for Ric' > 0 near Bd(D) is that the 'tangential projection' of h should be negative definite near Bd(D). The presence of the smoothing function ~ means that the formula for Ric' the terms in ~ will donfinate the therms in 02 and ~3 near Bd(D). This is precisely what makes R io '>0 near Bd(D) since the tangential projection of - h is essentially the leading term in ~ in the formula for Ric'.

Or we might say that local convex deformations work for the Ricci curvature because Ricci curvature is the trace of the sectional curvature. The deformation we use in Sections 4 and 5 has the property that for the first derivative K' of the sectional curvature

and

K ' (x ,V~) > 0 if x(e) = 0

K' (x, y) < 0 if x(e) = Y(O = O.

But the order in 0 of the contribution to Ric' from the 'radial two plane' {x, VO} is lower than the order in ~ of the sum of the n - 2 tangential two planes {x, y} in Ric'. Also since we are essentially taking the trace of K ' in computing Ric', we always have a radial two plane contributing a positive lowest order term to Ric'.

In Section 5 we use the results of Sections 3 and 4 to prove various Ricci curvature deformation theorems. We emphasize that since g(t) =go in M - D, we only have R i c ' > 0 on an open set, so some care is needed to be sure that we can find a t > 0 such that rict(v)> 0 for all v near Bd(D) simultane- ously. We apply one of the Ricci curvature deformation theorems to show that if a noncompact manifold admits a complete metric of nonnegative Ricci curvature and all Ricci curvatures positive at some point, then M has at most one end. We note that while this result follows immediately from the splitting theorem of Cheeger and Gromoll, [8], pp. 120, the proof of their splitting theorem is deep involving an application of potential theory. We thus offer a different and simpler way to understand this result on the end structure.

For completeness we remark that in [12a] we show that in general for dimension M > 3, there do not exist any local convex deformations for the sectional curvature positive at first order. Essentially we just turn the argu- ment of Section 4 around. The failure to find a local convex deformation g(t) of go with K ' > 0 near Bd(D) is a consequence of the following two statements:

METRIC DEFORMATIONS OF C U R V A T U R E , I 3

(i) the convexity of D implies that 6" (do) < 0, and (ii) since dimension M _ 3, we have two families of 2-planes, the infinite

family of radial 2-planes {x, V0} with x(0)=0 and the tangential 2-planes {x, y} with x(Q) =y(o) = 0. Then K' (x, VO) > 0 forces K'(x, y) >0 and vice versa.

To avoid any possible confusion, we will make explicit the following conventions we will use in this paper. We say that a func t ionf i s positive if f > 0 (in lieu of strictly positive), negative if f < 0 (in lieu of strictly negative), nonnegative if f > 0 , and nonpositive if f < 0.

I would like to thank Jean-Pierre Bourguignon for sharing his ideas on metric deformations stemming from his own work, [5] and [6], on the Hopf conjecture and for checking the calculations of Sections 4 and 6. I would like also to thank the referee for his constructive suggestions for improving the exposition of this paper and for correcting several minor errors in this paper and in 'Metric Deformations of Curvature II'.

2. THE GENERAL THEORY OF LOCAL CONVEX

DEFORMATIONS

Let M n be a C oo manifold, n>2 . Let TM be the tangent bundle of M and let T*M be the cotangent bundle of M. Let G2(M)-~M be the Grassman bundle with fiber at p all two dimensional vector subspaces of Mp. We will call these subspaces two planes as usual. Let R(M) be the convex cone of all Riemanlfian metrics for M. There is a natural action R(M) × Diff(M)--,R(M) given by (g,f)i-~f*g where (f*g)(v, w)=g(f,v,f,w). The quotient space of this action ~(M)=R(M)/Diff(M) is called the space of Riemannian struc- tures on M. (See [9].)

Given a Riemannian metric go on M, let D denote the Levi-Civita connection, Roo the curvature tensor, Koo:G2(M)~R the sectional curvature function, Ri%o the Ricci tensor, ri%o the Ricci curvature, and %0 the scalar curvature function determined by go. When there is no danger of confusion, we will write ( , ) for go and omit the subscript 'go' in the curvature tensors and functions determined by go. For definitions and sign conventions for R, K, Ric, ric, and z we follow [13], Sections 2.2, 3.5, and 3.6.

Let g(t) be a 1-parameter family of metrics with g(0)--go defined for t in some open interval about 0. We will write D t, R t, K t, Ric t, ric t, and z t for the operators, tensors, and functions determined by g(t). If we fix x, y~Mp, then t~Kt(x, y) is a real valued function so we can define

(1) K' (x, y) = d Kt (x, y) [ = lim K~ (x, y) - K(x, y) dt t=o t-*o t

4 P A U L EHRLICH

if the limit exists. Similarly we can define Ric'=(d/dt)Rictlt=o and ric' = (d/dt)ric'l,=o.

Given a Riemannian metric g on M, g defines an isomorphism of the tangent and cotangent bundles TM ~- T*M which on the fibers is given by M~v--*g(v,-)~M* where g(v,-)(w):=g(v, w). If ~C~(T*M) is a 1-form, then the vector field associated to ~ by this isomorphism which we will denote by ~% is given by g(~%, X)=~(X) for all X~COO(TM). Given a vector field X, there is a 1-form Xbg(Y)=g(X, Y). When there is no danger of confusion, we will write ~ for ~% and X b for Xbg. Let S2(M) be the bundle of symmetric two-tensors on M. There is a differential operator 0*:C °° (T*M)~ C OO (S2(M)) depending on g given by

(2) ~*~ (x, y) = ½L¢~og (x, y) = 2! ((Dx~) (y) + (D,~) (x)),

where D is the Levi-Civita connection determined by g and L is the Lie derivative. (See [2].) Properly, we should write ~* instead of ~*. But given a variation g(t) with g(0)=go, ~* will always be the operator defined by go so our notation will not be ambiguous. Le t f :M~R be a smooth function. We will write Vf for the gradient vector field associated to f which is defined by g (Vf, X) = X(f) for any X~ C °~ (TM). Then ~* (df) (x, y) = g (BxVf, y) is the Hessian form o f f The LaPlacian Af o f f is A f= tr ~* (df). We have

(3) ~* (d(fg)) = f6* (dg) + g6* (df)+ 2dfo dg,

where df odg= ½(df ® dg + dg @ df), and

(4) ,~* (f~) = f~*~ + d f o ~,

where f, g:M-+R are C 2 functions and ~C~°(T*M). The importance of 6" in the theory of metric deformations stems from

the decomposition of Berger and Ebin, [2], for Coo (S2(M)). Let 6' =D*:Coo (S2(M))--+COO(T*M) be the adjoint of the Levi-Civita connection determined by g. Then

Coo (S2(M)) = ker ~' ® im ~*.

Let us call a deformation g(t) of go whose 1-jet is not in the image of ~* a 'geometric deformation'. Let ~ C OO (T'M) and let g(t) =g + tO*~ =g + (t/2) xL¢%g. Consider the variation g(t)=qo*_tg where q~,:M~M is the 1-parameter group generated by ~*'. Then (M, g(t)) ~-~ (M, g) is an isometry and g(t) determines the same coset in ~(M) for all t. This deformation g(0 can thus be thought of as a coordinate change in ~(M). We will later be interested in (M, g) with Ko>0. It is not hard to see that given (M, g) with Kg>_0, ifg(t)=g+tO*~, then K(P)=0 implies K'(P)=0.

METRIC DEFORMATIONS OF CURVATURE, I 5

In [5] Bourguignon et al. studied for a fixed metric go for M arbitrary variations

t 2 h 2 t k g(t) =go + thl + - - + ... + - - h k with h i E C ~°

2 k!

× (S~(M)).

They defined a differential operator S : C ~° (S2(M))--~ C c° (S 2 (A 2 ( T ' M ) ) ) depending on go as follows. Given heC®(S2(M)), let g(t)=go+th and define (~h)(x, y)=(d/dt)Rt(x , y, y, x)lt=0. Then it is classical that

(5) (~h) (x, y) = DDh (x, y, x, y) - ~DDh (x, x, y, y)

- ½DDh(y , y , x , x ) + h ( R ( x , y ) y , x ) .

Suppose Ko_0 on M. Let P be a zero two plane with orthonormal basis {x, y}. Then if K(x, y )=0 , R(x, y ) y = R ( y , x ) x = 0 (see [3]). Hence i fx and y are a go-orthonormal basis for a zero two plane of Kgo, for any tensor he C ~ (S2(M)), the last term in formula (5) vanishes. I f g(t) =go + th ~ + ' " , then if x and y are go-orthonormal

(6) K ' ( x , y ) = ( S , h l ) ( x , y ) - K ( x , y ) ( h l ( x , x ) h ~ ( y , y )

- (h 1 (x , y ) ) 2 ) .

Thus, if K(x, y)=0 , then K' (x, y)=(~h~)(x, y). Let us fix some more notation for (M, g) once and for all. Put

Bg, R(p) := {q ~ M; dist o (p, q) < R} and

Ao, R. , (p) := {q~Bo,~(p); (1 -~7)R _< disto (p, q) < R}.

We will call Ao. ~, ~(p) the g-outer annulus of g-width ~TR for Bg, R(P). Let

r., p(q) := distg (p, q) = the g-distance from p and

~o(q) = Og,:c,p(q) = R - ro.p(q) = the g-distance

from Bd(Bg, 1¢(P)) which we will define only for q in Bo, g(p).

CONVENTION. All deformations g(t) of a given metric go for M will be at least C 3 in t through at least C A metrics for M.

DEFINITION. We will say that a set D contained in (M, g) is g-convex (or just convex when it is clear what metric we mean) iff for all p, q in D, there is exactly one normal minimal geodesic in D from p to q.

6 PAUL EHRLICH

The theorem stated in [1] referred to above suggests that a possible method of proving the analogue for sectional curvature would be to find a deformation g(t) of go with g(t)=go off a small disk D centered at the point Po of everywhere positive sectional curvature which would spread the positive curvature from a slightly smaller disk D' centered at Po to the annulus A : = D - D ' . The most obvious way to consider whether a geometric deformation g(t) is a 'good' deformation is to compute K'.

More generally, let D be a 'nice' connected open set in M with D compact so that if

o~ : D -~ R

is given by

0(q) := distg (q, Bd (D))

then gradg0 is smooth in some one-sided tubular neighborhood U c D of Bd(D). We will say t~g(t) with g(O)=go,t in ( - c , c), e > 0 is a local deformation o f go with support in D iff

(1) for all t in ( - c , c), g(t) = go in M - D,

(2) for all p in D, there exists v ~ 0 in Mp such that

g(t) (v, v) ~ go (v, v)

for all t ~ 0. From [5], in order to compute K' or Ric' for a deformation g(t) it is

enough to know the 1-jet ofg(t). Thus, to study K' or Ric' for an arbitrary local deformation of go with support in D we may assume

g(t) = go + th

for some symmetric two tensor h. Then the conditions g(t)=go in M - D and all metrics g(t) are C 4 imply that in order to study K' or Ric' near Bd(D) for an arbitrary allowable local deformation with support in D, it is enough to study all deformations of the form

g(t) = go + to 3h.

Given a two-plane Pc G2(M) with ~r(P) ~ U we will always choose a go-orthonormal basis {x, y} for P with x(0)=0. Then it follows from Corollary 4 given below that

K'(P) = - 30 (y(0)) 2 h (x, x) + 30z [20* (do) (x, y) h (x, y)

- ~* (do) (x, x) h (y , y ) - 6" (do) (y, y) h (x, x)]

+ ~2y (~) ((D~h) (x, y) - (D~h) (x, x)) + o 3 (Sh) (x,y).

METRIC DEFORMATIONS OF CURVATURE~ I 7

The presence of the Hessian 6" (do) suggests we choose D to be a go-convex disk so that 6" (do) will have a definite sign, Thus, we will study local convex deformations, that is, local deformations with support in convex metric disks

Boo, R(p). The formula above already shows the difficulty in using local convex

deformations to improve sectional curvature. Even though it is always possible to choose a tensor h on D=B,o.R(p ) so that (Xh)(x,y)>O i f {x,y} is a go-orthonormal basis for a two-plane P, Zh only enters the formula for K'(P) in third order in 0 and hence, near Bd(D) does not control K'. We remark here that a universal choice of a tensor h with Zh > 0 is

h = d & ) @ d(r:) ,

where r=rgo, p. The convexity of D implies that 6*(d(r2)) is positive definite on D = Bgo, R(P) and hence

(Z~h) (x, y) = (6*dr ~) (x, x) " 6" (dr ~) (y, y)

- (6* (dr ~) (x, y))~ > 0

by the generalized Cauchy-Schwarz inequality for positive operators. Geo- metrically, we can picture this deformation as follows. Let (M, go) := (R 2, g¢,n) with p = (0, 0). Then the metric

g(t) = go + td (r 2) ® d(r 2)

on R 2 can be represented by the metric on the paraboloid of revolution

M t : z = z ( t ) = t ( x 2 + y2)

induced from the standard metric on R a by the inclusion Mt c R 3. In later chapters and in [12a] with this discussion as motivation, we will

consider the following two problems:

PROBLEM I. Given D = Boo , x(p) convex, what are the possible local convex deformations of go with support in D and with Ric' > 0 in an annular neighborhood in D of Bd(D)?

PROBLEM IL Given D = Boo , R(P) convex, can we find a local convex deformation of go with support in D so that K ' > 0 in an annular neighborhood in D of Bd(D)?

8 PAUL EHRLICH

CONVENTION. Given ~ e C °O (T*(M)), X, Ye C ~ (TM), we will define

d~ (X, Y) = S . ~(r ) - Y .~(X) - ~ (IX, YI)

omitting the factor of one-half.

DEFINITION. Given U c D ~ M as above, p e U - B d D . For xeMp the

radial component of x written x o is Xo=go(X, Ve)Vo and the tangential component is x r = x - x o. We will say x is radial if xr=O and x is tangential if

Xo=O. This definition is motivated by the following geometric model. Let D be

a convex disk centered at Po. Then VO is the inward pointing normal vector field to Soo" Ro(Po)cD and Ve points in the 'radial' direction towards Po. A tangential vector lies in (V0) ± and is tangent to the sphere Sgo, Ro(Po).

NOTATIONAL CONVENTION. In the computational lemmas to follow, we will write (, } for go and D for the Levi-Civita connection determined by go. Given linearly independent vectors x, y sMp, extending to constant vector fields on Mp and composing with expp,, we get local vector fields X and Yin a neighborhood V o f p so that [II, Y]=0 in V and DXip=O and DYIp=O where DX(v):= D~Xip. We will call X and Y a good extension o f x and y. For a good extension,

(5') (_rh) (x, y) = x ~ (h (X, Y)) - ½xX (h (Y, Y))

- ½yY (h (X, X) ) - h (R (x, y ) y , x ) .

We now state a series of computational lemmas that we will use later. These are most conveniently derived using (3), (4), (5), and (5').

LEMMA 3. Let f : M ~ R be a C 2 function and h ~ C OO (S2(M)). Then

(Z fh) (x, y) = (3* (df) (x, y) h (x, y) - ½(3* (df) (x, x) h (y, y)

- } ( 3 * (df) (y, y) h (x, x)

+ x ( f ) ((D,h) (x, y) - (Dxh) (y, y))

+ y ( f ) ((Dj~) (x, y) - (Dyh) (x, x) + f(Zh) (x, y). Now

(3* (do") (x, y) = <D~V (e"), Y> = <D~ (ne"-lVo), y>

= n (n - l) e"-2x (e) y(e) + no "-~ (3* (do) (x, y) s o

(7) (3* (de") = no "-1 (3* (dO - n (n - 1) e "-2 de o dO.

M E T R I C D E F O R M A T I O N S OF C U R V A T U R E , I

In particular, i f xo=O, then

(8) a* (do") (x, y) = no "-~ ~* (do) (x, y).

Hence, we obtain

C O R O L L A R Y 4. For n >_ 3 in U

(Zo°h) (x, y) = n (n - 1) {2x (o) Y(0) h (x, y) - ]lXo][ 2 h (y, y) 2

- [lyol[ 2 h (x, x)} C -z

n 0,_ ~ {26* (do) (x, y) h (x, y) + - - 2

- a* (de) (x, x) h (y, y) - 6" (do) (y, y) h (x, x)}

+ no"-~y (~) ((D~h) (x, y) -- (D/O (x, x))

+ no"-~x (0 ((D~h) (x, y) - (D~h) (y, y))

+ O" (~h) (x, y).

Let ~:, ~C°~(T*M) and consider the variation g(t)=go+t~o ~. An elementary but lengthy calculation shows:

L E M M A 5

(~ (~ o 7)) (x, y) = ½7 (x) (D,d$) (x, y) + ~ (x) (DJv) (x, y)

+ ½~7 (Y) (Dxd$) (y, x) + ½~ (y) (DJ~) (y, x)

+ ½6"~ (x, x) O*~ (y, y) + ½O*~ (x, x)

x a*8(y ,y ) - ½dS(x ,y )a~(x ,y )

-- ½ (Dx~) (y) (Dx~) (y)

-- ½ (Dye) (x) (Dye) (x).

COROLLARY 6. I f ~ is closed, i.e., d~=O, then

( z (~ o 7)) (x, y) = ~ (,~(y) (DJ~) (y, x) + ~(x) (Dfl~) (x, y)

+ a*~ (x, x) a*n (y, y)

+ a*~ (x, x) a*~ (y, y)

- 2a*£ (x, y) ~*~ (x, y)).

10 PAUL EHRLICH

LEMMA 7

(S (03d0 o O) (x, y) = ~0 = (llxoll 2 6"~ (y, y) + Ilyoll 2 ~*~ (x, x~

- 2 ~ o d~o (x, y) 6"~: (x, y))

O 3 + - - (Y(O) (D~dO (y, x)

2

+ x(O) (Drd$) (x, y)

+ 6"~ (x, x) 6" (do) (y, y)

+ 6*$ (y, y) 6" (do) (x, x)

- 26*$ (x, y) d* (do) (x, y)).

COROLLARY 8. l f x and y are go-orthonormal with x o =0 and

g(t) = go + t~ado°$, then

O 3 K' (x, y) = -~0 2 [lyoll z ~'8 (x, x) + ~ (6*$ (x, x) ~* (d0) (y, y)

+ d*~ (y, y) 6" (do) (x, x) - 26"~ (x, y) d* (do) (y,x)

+ (y , VO) (D~d~) (y, x)) - 06 llY0112 (~(x)) 2 K(x ,y ) .

3, SOME REMARKS ON THE SPACE OF RIEMANNIAN METRICS

OF A SMOOTH MANIFOLD

The purpose of this section is twofold. First we make explicit the notion that if two metrics for a smooth manifold are close, then metric balls in the two metrics centered at the same point of the same radius are similar. This we use in Section 5 to prove the Pdcci curvature deformation theorems stated in [12].

For M compact, define the C 2 topology for R(M) by fixing a finite number of coordinate neighborhoods B~ and declaring [go-gllc:<6 iff for each B~, the coefficients of go and gl with respect to the fixed coordinate system on Bt and all their first and second derivatives are 6 close at each point in B~.

Let M be non-compact and let go~R(M) be complete. Recall that if a complete metric for a non-compact manifold is changed only on a compact subset, the resulting metric is also complete. Thus if K is a compact subset of M, then

FK. oo(M): = (g ~ R(M); g = go on TMIM.~,t,K) )


is a family of complete metrics for M. By compactness of K, it is clear that C 2 closeness in the sup norm for the family Ft:, oo(M) defined as for M compact still makes sense.

Assuming M is compact, we say a functional F:R(M)~R is C 2 locally minorized iff given goER(M), there exist constants 0(go)>0 and e(go)>0 such that if g~R(M) is O(go) close to g in the C 2 topology on R(M), then F(g)>_e(go). If M is non-compact, it is clear how to formulate a similar definition for families of metrics of the form Fr, go(M) where in this case the constants depend on K as well as go.

We apply the results of [11] and the comparison theory in Riemannian geometry to obtain the C 2 local minorization of the convexity and injectivity radius functions on R(M) for M compact and for families of metrics of the form F~, oo(11/1) for M non-compact.

For convenience all results of this section are formulated for smooth metrics. But the theorems of this section are evidently true for C 2 metrics and can thus be applied in Section 5.

Given two metrics gl and gz for a manifold M, we write Agl <_g2<_Bgl for constants A, B e R iff for all veTM, Agl (v, v)<_g2(v, v)<_Bgl (v, v). Re- call that we define a distance function disto:(M , g)x (M, g )~R from the Riemannian metric g for TM by

dist o (p, q) : = inf {Lo(c ); c is a sectionally smooth path from

p to q}.

(Here Lo(e ) denotes the arc length of c calculated with the Riemannian metric g.)

Given a go-outer annulus A=Aoo" R,~/4(P) defined as in Section 2, with e sufficiently small, it is reasonable that if g is another metric sufficiently close to go then A will be contained in a g-ball/~ centered at p of g-radius

slightly larger than R such that the g-outer annulus .~: =Ao, R,~(p) of B of width ei~ will contain A.

More precisely, just using the definitions of A and X, an elementary calculation shows

LEMMA 1. Given M and a fixed metric go for M. Fix e with 0 < e <_ ½. There exists a constant ~ > 0 with the following property. Let g be any metric for M with

(1 - 0)2go _< g < (1 + 6)2go.

Then for any p in M and R > O, there exists an R > 0 such that

Aoo,~,¢/4(p) = Ao, R,¢(p).


We make some standard definitions. Given (M, g) complete, let io(p) :=sup(R>0; expp:Uo, R(p).-.Bo,~(p ) is a diffeomorphism} called the in- jecfivity radius of (M, g) at p and f o r / £ = M let

io, K(M) : = inf {io(p); p ~ K}

called the K-injectivity radius of (M, g). Let io(M): = in, M(M). We say C¢(M, g) is g-convex iff for all p, q~C there is exactly one nor-

mal minimal geodesic in C from p to q. We define

ca(p) := sup {R > 0; B,,R(p) is g-convex}

which is the convexity radius of (M, g) at p, and for K= M let

eo, K(M) := inf {co(p); p ~ K}

and put co(M): = co, u(m). Let

L(g) := inf {Lo(c); c is a smooth closed non-trivial g geodesic}.

It is well known that for fixed complete g, the maps pw, io(p) and p~-+co(p) from M ~ R are continuous functions.

We want to Consider the local behavior of g~-->io(M ) and g~-->co(M) as functions R(M)~R on the space of metrics for M. Fixing a Riemannian metric go for M, recall that ioo(p ) and Coo(P) are determined by the behavior of the configuration of radial geodesics from p. For g close to go in the C z topology on R(M), the configuration of g-radial at p is close to the go-configuration of radial geodesics at p by the theory of ordinary differential equations (see [11]) and thus these functionals should be locally minorized.

From our proof in [11] of the lower semicontinuity of the map g~io(p) from R(M)~R with the C 2 topology on R(M) for M compact, it is immediate that

THEOREM 2. Let K = M be compact and goeR(M) be complete. Define F~, ,o(M) as above. Let gl eF~, ,o(M). Then there exist constants ~ (gl , K)> 0 and 1(gi, K) > 0 such that g2 ~FK, go(M) and [gl -g2 Ic2, ~ < ~ (gl, K) implies ig2, K(M) >I(gl , K). In particular if M is compact, then g~io(M ) is C 2 locally minorized on R(M).

Remark. It is an immediate consequence of Theorem 2.1 of Cheeger, [7], on families of Riemannian manifolds of bounded volume, diameter, and sectional curvature that for M compact, g~L(g) is C 2 locally minorized and hence g~ig(M) is C 2 locally minorized. However, Cheeger's result cannot be modified to apply to families of complete metrics F~, ,o(M) for M non- compact. Thus it is really necessary to study the configuration of radial

METRIC DEFORMATIONS OF CURVATURE, I 13

geodesics from a fixed point in M for all metrics in a neighborhood of a given metric (which we do in [11 ]) to obtain the local minorization in the noncompact case.

Now we turn our attention to the C z local minorization of the convexity radius function.

DEFINITION. Bo, R(p) is said to be g-good iff for all q in Bg, R(p) the exponential map

M~ = Uo,2R(q) ~ Bo,2R(q) = M

is a diffeomorphism. The following lemma from [13], p. 160, shows what is needed to minorize

the convexity radius function on R(M). We make the convention that for p~M, veMp with g(v, v)=l that we write

cv(0 := expp tv

for the g-radial geodesic from p in unit direction v.

LEMMA I. Suppose Ba, R(P) satisfies (A) Bo, R(P) is g-good, and (B) for all v~Mp with g(v, v)= 1 the index form Ico is positive definite on

all Jacobi fields J along cv with initial conditions J(0)=0 and g(cv, J)=0. Then Bg, ~(p) is g-convex.

Given (M ", go) complete and Kg o bounded from above, the proof of the Index Comparison Theorem, [13], p. 174, shows that the behavior of (B) of Lemma I is minorized for gC ~ close to go. Explicitly,

PROPOSITION 3. Given M compact and k>0. There is a constant R(k)>0 with the following property. Let g be any complete Riemannian metric for M with K o <_k. I f B is any g-good metric ball of g-radius < R(k), then B is g-convex.

Putting together Lemma I, Proposition 3, and Theorem 2 we obtain

THEOREM 4. (C 2 local minorization of the convexity radius function.) Given M compact and a Riemannian metric go for M, there exists constants 6(go) > 0 and C(go)>0 such that i f g rR(M) is 6(go) C 2 close to go, then any g-metric disk on M of g-radius < C(go) is g-convex.

For the Ricci curvature deformation theorem to be given later, we need an estimate on

~* (d(rg, p)),


where as above ro,~(q): = dista(p, q), or equivalently on the index form Icy of Lemma I, for all metrics g sufficiently close to a fixed metric go for M.

It is clear that the idea involved in Theorem 4, especially the idea of the proof of the Index Comparison Theorem, imply

THEOREM 5. Given M compact and a Riemannian metric go for M. There exist constants d(gG) > 0 and F(go) > 0 such that gE R(M) and I g-golc~ < O(go) implies any g-disk centered at any p ~ M of g-radius <F(go) is g-convex and

(2 - ¼) g <_ 6" (d((ro, p)2)) _< (2 + 9 g

holds in any such disk. Remark. As a motivation for this inequality recall that O*(d((ro, p)2))lp

=2gl~.

It is clear that Theorem 5 extends to families of metrics on the form FK.oo(M). Thus we state

THEOREM 6. Given M noncompact, K c M compact, and go~R(M) complete. Let FK, oo(M) be defined as above. Let gl EFK, oo(M). Then there exist constants ~ (gl, K) > 0 and F(gl, K) > 0 such that g~Fr, oo(M) and Ig-g , [c2,r <O(g~, K) implies that any g-metric disk centered at any p~K of g-radius < F(gl, K) is g-convex and

(2 - ¼)g < ~* (d((ro,,)=)) _< (2 + ¼)g

holds on any such g-metric disk.

4. C A L C U L A T I O N OF R I C ~ FOR L O C A L C O N V E X D E F O R M A T I O N S :

THE S O L U T I O N OF P R O B L E M I

In this section, given (M, go) and p ~ M we consider

PROBLEM I. Given D : = B,o,a(p ) convex, what are the possible geometric deformations g(t) of goC 3 in t through C 4 metrics with support in D and with Ric '> 0 in an annular neighborhood of Bd(D) in D ?

and show that this problem can be solved in particular by a conformal deformation.

Recall that technically g(t) geometric means that if h is the 1-jet of g(t), then h~Imd*. Also in Section 2 we saw that it was enough to study deformations of the form g(t)=go + toah to compute Ric' for all possible deformations satisfying the conditions of Problem I.

METRIC D E F O R M A T I O N S OF C U R V A T U R E , I 15

F i x p s M . Remember that i f r is the distance f romp on M, then in M - {p}, r is a smooth function up to the cut locus o f p and r 2 is a smooth function on M up to the cut locus ofp . We may take U: = D - {p} as the one-sided tubular neighborhood of D discussed earlier. Then the distance function 0 to Bd(D) is just 0 = R - r . In D - { p } , go(Vr, Vr)= 1 so that

O* (dr) (x, Vr) = go (D~Vr, Vr) = x (go (Vr, V0)/2 = x(1)/2 = 0

and hence we obtain

(9) a* (de) (x, y) = d* (de) (XT, Yr),

where Xr is the tangential component of x defined as in Section 2. Given heC~(S2(T*(D))), we may write h=hr+doo~+fdoodo where

f :D-+R is a smooth function, ~eC®(T*(D)) with ~7(Vo)=0 and hr(x,y) =hr(Xr, Yr). We will call hr the tangential component of h. For now we combine d~ o,~ +fdo~ o a'~ as d o o ~ where ~ e C ~ (T* (D)).

It is straightforward to calculate Rie'(v, v) for h=hT+dOo~ but all we need here is the first order terms. Explicitly

LEMMA 2. For go (v, v) = 1 and g(O =go + t03 (hr + d~ o ~),

Ric' (v, v) = - 3 0 [hr (v, v) + Ilv0[i 2 t r h r ] + O(oZ).

From Lemma 2, if hr(v,v)<_-kgo(v, v) with k > 0 a constant, near Bd(D), then Ric' will be positive in an annular neighborhood of Bd(D) in D. Thus the answer to Problem I is YES. The most obvious choice o f h r satisfying this condition is h r : = - g o ! (V0) ±.

This leads us to consider the deformation gl ( t )=go- tOago . However the conformal deformation g~(0 = e-~°~ go has the same 1-jet as g~(t) and hence the same first time derivative of the Ricci curvature. Since the formulas for the Levi Civita connection and curvature tensor for the conformal deformation are simple and explicitly known for all t, we will use ~l(t) to perturb the Ricci curvature instead of gt(t).

We now consider the conformal variation g(t)= e-Zt°Sgo on convex disks. I f g(t)= eZtfgo, then it is well-known that

tic t (v) = e -2~s {tic (v) - t (n - 2) ~* (df) (v, v) - t a r

+ t 2 (n - 2) ((v(f)) 2 - HVflt2)}.

Hence for f = - 0 s, Ilv][ = 1,

tic r (v) = e 2to5 {fie (v) + t (n - 2) 6* (d05) (v, v) + t A(0 s)

+ t ~ (n - 2 ) ( (~(o~)) ~ - [IV(~o~)ll~)}.


But ~* (d05) (v, v) = 2093 IIvoll = + 594~* (rig) (v, v),

Hence

Thus,

A(95)=2003 + 55"A 9, and

(v(95)) z - []V(95)I[ z = -2558 Ilvrll z

ric' (v) = e z'°s {ric (v) + 20@ (1 + (n -- 2) ilv01l 2)

+ 5t94 (A 9 + (n - 2) b* (de) (v, v))

- 25t 2 (~ - 2)9 ~ liVTII2}.

rict (v) _> e 2t~ {ric (v) + 20t03

+ 5t9" (A9 + (n - 2) ~* (de) (v, v))

- 25 (n - 2) t29s}.

Let p ~ (M, go) be a point where all Ricci curvatures are positive (assuming riCoo_>0). Thus, in some small closed disk about p all Ricci curvatures are positive. We want to change the metric so that all Ricci curvatures will be positive on a larger convex disk D = Bgo, R(P) of radius R and so that the new metric agrees with go off D.

I f we let r be the go-distance from p on M, then 9 = R - r so d*(do) =-c3*(dr) and A g = - A t . By choice of R according to our convention, O* (de) will be negative definite on D - {p}. At p, the function r 2 is smooth and d*(d(rZ))(v, v)=2go(v, v) for all v in Mp, as we showed in Section 3. We may then by continuity and/or the Index Comparison Theorem tech- nique of Section 3 choose R small enough so that on D

(9) (2 - ¼) go --- c~* (d(r2)) <_ (2 + ¼) go.

Hence on D - { p } we have

(1 - 1 / 8 ) 1 + 1/8 (9') go (xr , xz) _< ~* (dr) (x, x) < go (xr , xr) .

r v

Remember that R=r+9. We will assume R _ I so that 5___1. From (9), 10*(d0)(v, v)l <91JVTII2/8(R-9)<9/8(R-9) so IAgl < 9 ( n - 1) /8(R-9) . (Re- member that 6 ' (do)(V9, V9)=0.) Thus

9 (2n - 3) tAo + (n - 2) ~* (do) (v, v)l <

8 ( R - 5)


so ric t (v) > e 2t°' {ric (v) + 5t03 [ 4 - 9 ( 2 n - 3)~o/8 ( R - ~o) + 5 ( n - 2) t0s]}. By

choosing t < 2/5 ( n - 2 ) in D - { p } we obtain

(10) tic ~ (v) > ric (v)(1 + t0 s) + 5t~o a [2 9 (2n - 3) ] - 8 ( g 0 ) 0 •

Now if 2 - [ 9 ( 2 n - 3 ) / 8 ( R - o ) ] o > O and 0 < o < R , then r ict(v)>0 for all v. But this is true if 0 < ~ < 1 / (1 + 9 (2n - 3)) R. Hence, we have shown

T H E O R E M 3. There exists a constant e=e(n)>O with the following property. Given (M", go) with Ri%o>0, let D=Boo,R(p ) be any disk for which (9) holds and R <_ 1. Let

g(t) = f ~ e-2t°~g° in D

[go in M - D.

Then there exists to > 0 such that for all t e (0, to], r ict> 0 on the go-outer annulus Aoo ' R,z~n)(P).

Remark. We need to consider rict instead of merely ric' since ric' (v)> 0 only on some open annulus about Bd(D) (since g(t)=go on Bd(D)). From ric' (v)> 0 on an open set we cannot conclude that there exists a small t such that r ict(v)>0 everywhere on the open set even though for each point q in the open set from the positivity of ric' at q we can find a t(q) so that r i d ( ° ( v ) > 0 for all v~M o. Of course if r i c '>0 on a compact set, we can find a t > 0 so that r ict> 0 everywhere on that set.

The analogue to Theorem 3 holds for ricoo < 0 using the deformation

g(t) = f ~ e2t°Sg° in D

[go in M - D.

Remark. In Theorem 3 since 0 = R - r is not smooth at p, the deformation given is not smooth at p. But we are only interested in the positivity of the Ricci curvature in the outer annulus in Theorem 3. Clearly given D = Bo, a(P), g(t) can be smoothed off near p to produce a metric satisfying the conclu- sions of Theorem 3. We will call this modified deformation the 'standard deformation of Theorem 4.3' in the next section.

ADDENDA. Taking g(t)=e2t¢g o with f = -O k (resp. f = - e -°2) gives a standard deformation of g through C k (resp. C °°) Riemannian metrics to produce rict> 0 in a uniform go-outer annulus as in Proposition 3 above. Taking f = 0 k (resp. f = e -Q2) produces a standard deformation for negative Ricci curvature in a uniform outer annulus for C k (resp. Coo) Riemannian metrics. The calculations are analogous to those of Theorem 3.

18 PAUL EHRLICH

5. PROOF OF THE RICCI CURVATURE DEFORMATION THEOREMS

In this section we use local convex deformations to prove several theorems on deformation of Ricci curvature. We prove for 4 _ k_<

THEOREM 1. Suppose M", n_>2, admits a complete C k metric go with riCoo > 0 (resp. riCoo_< 0) and all Rieei curvatures positive (resp. negative) at some point. Then M admits a complete C k metric o f everywhere pos#ive (resp. negative) Ricei curvature.

Proof. We discuss only the case rio>0. The case tic<_.0 is handled the same way except that the deformation g(t)=e2tOSgo is substituted for the deformation g( t) = e- 2t°Sgo , etc.

The basic idea to prove Theorem 1 is to use the standard deformation of Theorem 4.3 to spread the positive Ricci curvature from the point of positive curvature to all of M.

We first assume M is compact! Explicitly Theorem 4.3 tells us that given positive Ricci curvature on a

small enough disk, we can spread the positive Ricci curvature to a slightly larger disk centered at the same point, provided the larger disk in convex in the original metric.

But if we just start changing the metric naively, even though we can always find a convex disk in which to apply Theorem 4.3, it is not obvious that the radii of the disks in the original metric can be chosen to remain bounded away from zero so that we can cover all of M with even an infinite number of deformations. In particular, the convexity radius is changing with each deformation.

However, we can use the compactness of M to overcome this difficulty B as follows. Cover M by N balls {B~ = oo.R~(P~)}~= t with R~ <_ F(go)/2, F(go)

as in Theorem 3.5, satisfying the following properties. Let e=e(n) be as in Theorem 4.3.

(1) B~ is chosen so that p~ is a point with all go-Ricci curvatures positive in B~--Aoo,~l.~/g(pl ) and thus the standard deformation of Theorem 4.3 appfied to B~ will produce a metric gt for M with ricg~ _ 0 and ric~l > 0 in B~.

(2) Inductively for n_> 2, if ricao were positive on

B 1 u B 2 u . . . k) Bn_ 1 u ( B n - Aao, R.,~/4(Pn))

then the standard deformation of Theorem 4.3 applied to B n would produce a metric g for M with rica>0 and ricg>0 on B~w...wB~.

METRIC DEFORMATIONS OF C U R V A T U R E , I 19

(3) the go-outer annuli N {Aao, R,,e/4(pi)}i= 1 cover all points q in M for which there is a v # 0 in M~ with ri%o(V)=0.

For instance, assume tl%o>0 in Boo,R(p ) to begin with. We can choose a finite number of balls as required to cover all possible points with zero Ricci curvatures in Boo,(I+,/s)R(P). In fact, it is clear that for all n, with a finite number of balls we can extend the cover of all possible 'zero points' from Boo,(~+,~/8)R(p ) to Boo,(l+(,+~)~/8)R(p ). Since M is compact, we can thus

B N clearly produce the required sequence { t}~=~ stipulated above. Let 0(go) satisfy Theorem 3.5 and Lemma 3.1. Use the standard deforma-

tion of Theorem 4.3 to produce from go a metric g~ for M with

[go - g11c2 < ~(go)/N

with rico1 > 0 on M, and rico 1 > 0 on B1 (which is possible since t lct> 0 on Aoo,Ri,2,(pl) by Theorem 4.3).

By Lemma 3.1 applied to g l , we can find a disk B2=Bo,,~(p2) which is g,-convex and with

A = Aoo ' R~, ~/4(P2) ~ Ao~, ~ , ~(P2) = -~

and /~ < F(go).

Hence we can apply the standard deformation of Theorem 4.3 to g~ and /~z to produce a metric gz for M with

lgl - g ~ l ~ < ~(go)/N,

rico2 > 0 and rico~ >0 on Ba w/~2 and hence on B~ wB2, spreading the positive Ricci curvature to Aoo" R~/4fPz). Note that by construction

Jgo -g_~J~ < ~(go).

Carrying on this way we construct mettles g3,...,gN applying the standard deformation to balls B3 = B3,... ,/~N ~ BN to spread the positive Ricci curvature from go-outer annulus to go-outer annulus making each new metric g, O(go)/N close to the preceeding metric g ,_ , and hence 0(go) close to go so Lemma 3.1 and Theorem 3.5 apply to enable us to carry out the next step.

Q.E.D. in compact case. Given (M, g), let Boo , ~[p]: = {q~M; distoo (p, q) < ~}. Substituting Theo-

rem 3.6 for Theorem 3.5 and F(go, K) for F(go), our proof of Theorem 1 in the compact case shows

PROPOSITION. Let M ~ be non compact and let K c M be compact. Sup- pose go is a complete C 4 metric for M with nonnegative Ricci curvature and


some point p ~ K with all Ricci curvatures positive at p. Let Boo,o(K ) := uq~KBoo,~[p]. Then go can be perturbed by a finite number of standard local convex deformations to a complete C 4 metric g for M with everywhere nonnegative Rieci curvature and all Rieei curvatures positive on K. Further- more, given ~ > O, go may be perturbed to such a metric g so that go is changed only on the neighborhood Boo,o(K ) of K.

Suppose M", n_> 2, is non-compact and admits a complete C 4 metric with nonnegative Ricci curvature and all Ricci curvatures positive at some pointp. For all positive integers m, let B,n : = B o o , m [P] and K m ". = B,. - Int (Bin_ 1) for m_> 2. Since go is complete, B,. and Km are compact. Using the proposition we may define a sequence of complete metrics gm with rico. >0 and rico. >0 on B,. and gin=g,._1 in TMIB,.+I-n,._~. That is, g,,~ is constructed from gin-~ by using the Proposition to spread the positive Ricci curvature from Bin_ ~ to Km changing gin-1 only in some neighborhood B0..(/£,.) for some small 0,~ choosen so that in particular

(*) B~m(K,,) = Bm +1 -- Bo,_I .

By (*) there are no convergence problems in producing a complete C 4 limit metric g with ric o > 0 everywhere.

Q.E.D. to Theorem 1 As a corollary of Theorem 1, we have

THEOREM 2. I f M", n > 2, is noneompact and admits a complete metric of nonnegative Rieei curvature and all Rieei curvatures positive at some point, then M is connected at infinity, that is, M has only one end.

Proof If M is not connected at infinity, then the standard classical construction produces a line for any complete metric on M. But Gromoll and Meyer in [14] remarked that a complete open manifold of positive Ricci curvature is connected at infinity.

Q.E.D. Suppose (M, go) is a compact Riemannian manifold with isometry group

Igo(M ). Suppose riCoo_ 0 and at some point rico° > 0. Then we could apply Theorem 1 to produce a metric g on M with rico>0. But if Igo(M) is not discrete, it is not obvious that the construction outlined in the proof of Theorem 1 will result in Ioo(M)c Ig(M). However, using the idea of Alan Weinstein in [17] to integrate the metric deformation of Theorem 1 over the isometry group it is possible to use local deformations to produce a new metric g with ric o > 0 and Ioo(M ) ~ Io(M ).

We want to consider whether local convex deformations can be used to improve Ricci pinching.

METRIC D E F O R M A T I O N S OF C U R V A T U R E , I 21

DEFINITION. (M", go) is positively Ricei pinched with pinching constant

0 < A < I iff for all v~TM, Akgo(V,v)<Ricoo(V,V)<_kgo(V,V ) for some k > 0. I f (M, go) is Ricci pinched, then multiplying the metric by a constant we may assume that k = 1.

Suppose (M", go) is a compact manifold with Ago < R i% o <go and there exists p ~ M such that the pinching is not attained for any veM~. By compactness arguments, we can find a closed convex diskD' centered at p of radius R<¼eao(M ) and 2 > 0 so that for all v~SI(M, gO)ID either

(11) A + 2 < R i c ( v , v ) < 1, or

(12) A < Ric(v ,v) < l -- 2.

Let D be a disk centered at p with D ' c D. For the local convex deformation g(t) = e- 2t°Sg o on D,

Ric t (v, v) = Ric (v, v) + 20t93 (1 + (n -- 2) Ilvotl 2)

+ 5te" (Ae + (n - 2) 8" @ ) (v, v))

- 25t 2 (n - 2) 0Sltvrll 2

= Ric (v, v) + 20to 3 (1 + (n - 2) Ilvoll 2)

+ t o (e*) + t ~o (es).

Then, fixing q e M - {p}, maXoo,V ' ~, =l {Rict( v, v)} < 1 + 20t~ 3 (n - 1) + tO(o 4) veMq

+ t20(9 s) and minoo(~" ~)= ~ {Rict (v, v)} >A + 20t9 s + tO (~4) + t~O(eS). vsM~

As in the proof of Theorem 1, the term in ~ of order 3 will dominate the two higher order terms. The condition for improving the Ricci pinching is that the ratio

rain Rict]~ be greater than A. But we have max Rict[.

min Rictl, A + 20t~ 3 > + tO (~¢)

max Rict[~ 1 + 20t~3 (n - 1)

= (a + 20@) (1 - 20@ (n - 1)) + t20 (~6) + tO (~4)

= A +20 tp 3 ( 1 - A ( n - 1 ) ) + t O ( ~ 4 ) .

By choosing t and ~ small we can make this ratio greater than A if 1 - A (n - 1) >0, that is, if A < l / ( n - 1 ) . Hence if the annulus D - D ' is small and A < 1/(n- 1), we can improve the pinching in the annulus. It is clear that we can obtain a uniform estimate for the size of the annulus depending only

22 PAUL EHRLICH

on the dimension n of M as we did in proving Theorem 1. Also an easy calculation shows we can keep the pinching greater than A in D'. Hence we have

THEOR EM 3. Let (M n, go) be a compact Riemannian manifold. Suppose there exists k > 0 and A with O<A < l / ( n - 1 ) so that on T M the inequality Akgo <<_ Rico o < kgo holds. I f at some point for all non-zero vectors the pinching is not achieved, then it is possible to improve the Ricci pinching.

The analogous theorem holds for negative Ricei pinching using the local convex deformation g(t) = e2t°~o.

N. Hitchin, [15], has shown that there is a relation between the Euler class and the signature of a 4-manifold M 4 which is an obstruction to M 4 ad- mitring an Einstein metric. S.T. Yau, [18], generalized the result of Hitchin to show there is a similar obstruction to improving pinching of the Ricci curvature tensor. Yau gave an example of a compact negatively Ricci

curved manifold that does not admit a ~/~ Ricci pinched metric. For n = 4, the bound of our Theorem 3 is ½ so there is a gap between

the 'geometric' obstruction of Theorem 3 and Yau's 'topological obstruc-

tion' of ~/~Y. It is thus important to see whether the 'geometric' obstruction of 1/(n- 1) in improving Ricci pinching using conformal deformations in a convex disk can be improved by using some other type of local convex deformation.

It is possible to show, however, by studying the ratio of max(1 + tRic ' ) to rain ( A - t Ric') for an arbitrary local convex deformations g(t)= go + @ h (decomposing

h = h r +fd~oo~d~ + d~o~ where ~(AQ) = 0 )

that assuming only that the pinching does not hold at some point we cannot improve the bound of 1/(n-1) from conformal variation. Thus conformal deformation is the 'best' variation for improving Ricci pinching by the method of local convex deformations.

BIBLIOGRAPHY

I. Aubin, T., 'Metriques Riemannienes at Courbure', J. Diff. Geometry 4 (1970), 383--424. 2. Berger and Ebin, 'Some Decompositions of the Space of Symmetric Tensors on a

Riemannian Manifold', J. Diff. Geometry 3 (1961), 379-392. 3. Bishop and Goldberg, 'Some Implications of the Generalized Gauss-Bonnet Theorem',

Trans. Am. Math. Soe. 112 (1964), 508-535. 4. Bishop and O'Neill, 'Manifolds of Negative Curvature', Trans. Am. Math. Soe. 145

(1969), 1-49. 5. Bourguignon, Sentenac and Deschamps, 'Conjecture de H. Hopf sur les vari6t~s

produits', Ann. Sei. de L'Eeole Normale Superieure (1972), 277-302.


6. Bourguignon, Deschamps and Sentenac, 'Quelques variations particuli~res d'un produit de mrtriques', to appear in Ann. ScL de L'Eeole Normale Superieure.

7. Cheeger, 'Finiteness Theorems of Riemannian Manifolds', Am. Y. Math. XCII (1970), 61-74.

8. Cheeger and Gromoll, 'The Splitting Theorem for Manifolds of Nonnegative Ricci Curvature', J. Diff. Geometry 6 (1971), 119-128.

9. Ebin, 'The Manifold of RiemannJan Metrics', Proe. of the Symposia in Pure Math. o f the A.M.S., Vol. XV: Global Analysis (1970), 11-40.

10. Ehrlich, 'Complete Non-positively Curved NegativelyPointed Metrics on S ~ ×.S 1 × R2, preprint, Centre de Mathematiques de l'Ecole Polytechnique, Paris, France

11. Ehrlich, 'Continuity Properties of the Injectivity Radius Function', Compositio Mathematics 29 (1974), 151-178.

12. Ehrlich, 'Local Convex Deformations of Ricci and Sectional Curvature of Compact Manifolds', Proceedings o f the A.M.S. Symposium in Differential Geometry 27 (1975), 69-71.

12a. Ehrlich, 'Local Convex Deformations and Sectional Curvature', Arehiv der Math. 26 (1975), 432--435.

13. Gromoll, Klingenberg and Meyer, Riemannsche Geometric im Groflen, Springer- Verlag, Lecture Notes in Mathematics, No. 55, 1968.

14. Gromoll and Meyer, 'On Complete Open Manifolds of Positive Curvature', Ann. of Math. 90 (1969), 75-90.

15. Hitchin, 'On Compact Four-Dimensional Einstein Manifolds', J. Diff. Geometry 9 (1974), 435--443.

16. Preissman, 'Quelques Proprietes Globales des Espaces de Riemann', Comm. Math. Helv. 15 (1943), 175-216.

17. Weinstein, 'Positively Curved Deformations of Invariant Riemannian Metrics', Proc. Am. Math, Soe, 26 (1970), 151-152.

18. Yau, S.T., 'Curvature Restrictions on Four Manifolds', to appear.

Author's address:

Paul E. Ehr l ich ,

Ma thema t i s ches Ins t i tu t de r Universit~it Bonn

53 Bonn,

Wegele r s t r age 10,

F e d e r a l Repub l i c o f G e r m a n y

(Received April 17, 1974; in revised form February 1, 1975)

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