432 ARCH. MATH.

Local Convex Deformations and Sectional Curvature

By

PAUL E. Em~Lm~

By a local convex de]ormation of a Riemannian metric g for a manifold M we mean a 1-parameter family t~-> g(t) of Riemannian metrics with g ( 0 ) = g agreeing uni- formly in t with g off some g-convex metric disk. Here g-convex means every two points in the g-metric disk can be joined by a unique minimal normal g-geodesic lying in tha t disk.

We saw in [6] tha t given a g-convex metric disk D in (M, g) there exists a class of local convex deformations with support in D so tha t given any deformation t ~-> g (t) in this class, there exists some annular neighborhood of Bd (D) in D so tha t Ric' > 0 in this neighborhood. Here Ric ' denotes the first t ime derivative of the 1-parameter family of Ricci curvature tensors t ~-> Ric t determined by the metric deformation t ~-> g (t). In other words, there exists a local convex deformation of the Ricci curva- ture positive a t first order in an annular neighborhood of Bd (D) in D. One particular such local convex deformation can be used to deform a complete metric with non- negative Ricci curvature and all Ricci curvatures positive at some point to a complete metric of everywhere positive Ricci curvature (see [1], [6]).

In light of Gromoll and Meyer's construction, [7], of a Riemannian metric for an exotic sphere with nonnegative sectional curvature and all sectional curvatures positive at some point, the following conjecture has aroused recent interest.

Conjecture. Let (M n, go), n ~ 3, be a compact Riemannian manifold with all sectional curvatures nonnegative and suppose there is a point with all sectional curvatures positive at that point. Then M admits a Riemannian metric of every- where positive sectional curvature.

The purpose of this note is to point out the delicacy of this problem by remarking tha t in general no local convex deformation positive a t first order for the sectional curvature exists. The method of conformal deformations tha t yields the existence of local convex deformations positive at first order for the Ricci curvature cannot be generalized by using a more subtle deformation than a conformal deformation to produce a deformation of the sectional curvature on a convex disk D with positive first t ime derivative K' >0 near Bd(D).

This result should perhaps be compared with Berger's result, [2], on the H o p f conjecture for product metrics on S~ • S 2. Berger noticed that any deformation of

Vol. XXVI, 1975 Local Convex Deformations 433

a product Riemannian structure for S 2 • S 2 nonnegative at first order vanishes at first order. Both Berger's result and the result given in this note can be viewed as asserting the delicate nature of these two curvature problems on nonncgatively curved compact manifolds.

Let (M, go) satisfy the hypothesis of the conjecture. We use notations established in [6], section 2 and some results from [6]. In particular we need the formula given in [6], section 2, Corollary 4 which we will refer to as "formula (*)" below. Let

: G2 (M) -> M denote the Grassmann bundle of 2-planes in TM, let K : G2 (M) -> ]K denote the sectional curvature function of (M, go), and given a deformation t ~-~ g(t) of go, let K ' denote the first t ime derivative of the family t ~-> K t of sectional curvature functions defined by the metric deformation. Let Z (go) :---- K -1 (0) c G2 (M).

We consider C 3 variations in t ime and space parameters with support in a closed convex disk D c M {but D =~ M) with center 10 e M. (If D ~-- M is convex then, [3], a conformal deformation using the distance function from io will produce a metric of positive sectional curvature.) Let ~: D--> ]K denote the distance to Bd(D) as in [6] and following [6], for tangent vectors x lying over D -- {1o} let xe : = go(x, VQ)VQ and xT : = x - - xe.

Definition. A (C s) local convex deformation g (t) of go with support in D will be called positive at first order iff K' > 0 on Z (go) • ~-1 (Int (D)).

F rom now on, let (M n, go) be the 3-sphere S s flattened near the North pole so tha t if go is the Riemannian metric induced by the inclusion S s c (JR s, gcan) then K = - - Kgo -~ 0 on the closed ball N of radius 1/4 about the North pole and K ~ 0 in S s - - N . Let ~o e S s - - N be a point near N in the Northern hemisphere. Thus K > 0 on ~-1(10). Let D be any convex disk centered at 10 so tha t Bd(D) n / V ~ 0. In D n N all sectional curvatures are zero.

Theorem 1. There is no local convex deformation with sup10ort in D positive at first order.

We note tha t Theorem 1 is true with the analogous construction of N and D on the flattened n-sphere for all n ~ 4 and that the analogous observation is also true for non-positive sectional curvature with the same proof.

P r o o f o f T h e o r e m 1. Fix an arbi trary local convex deformation g(t) of go with support in D. Then from [6], section 2, there is a one-sided tubular neighborhood U of Bd (D) n / V in D so tha t we may assume tha t on U, g (t) has the form g (t) = go -t- t ~3 h, h smooth on U. (Remember we are only considering K'.) As in [6] decompose h = h~, ~- dQ o ~ as the sum of a "purely tangential" tensor hT satisfying h~ (x, y) = = hT(xT, y'2) and a "mixed tensor" d~ o ~.

We say a 1-form ~ is a tangential Killing 1-form on U1 c U iff

(~* V) (x, y) ---- (L~ go) (x, y)/2 ~- 0

for all tangent vectors lying over U1 with x (~) ---- y (Q) ~ 0, where L denotes the Lie derivative operator and ~ is the vector field associated to ~ by go- For brevity we refer below to tangent vectors x lying over D with x (~) ---- 0 as "tangential vectors" below.

Arehiv der Mathematik XXVI 2 8

434 P.E. EHRLICH ARCH. MATH.

Lemma 2. For K ' ~ 0 in In t (D) (~ N, we must have hT ---- 0 in some neighborhood U1 c U o / B d (D) ~ ~V.

P r o o f . I f not, then there exists a sequence ( p i ) i ~ c D so tha t hT ~= 0 at pi and lira dista0 (pi, Bd(D)(~ N)----0. Since hT is a symmetric 2-tensor, we may choose i - -~o g0-orthonormal "tangential vectors" xi and y~ in M~, so that h~ (xi, y~) ~ O. For all "tangential vectors" v we have the "radial" 2-plane (v, Ve). By construction, we have also K(x l , Yt) ---- K(v , Ve) ~ 0. :From formula (*) we have

K'(v, Ve) = - - 3~hT(v , v ) "t- O(e 2) (1)

and also (2) K'(xt , y~) : - - -~ e 2 [5* (de) (Yt , yi) hT (x~ , x~) Jr

~- 8" (de) (xi, xi) hT (y~, y~)] + 0 (e3).

From (1) for v : x /o r yi for i large, hT(x~, xi) ~ 0 or hT(yi , Yi) ~ O, and h•(v, v) ~ 0 for v = xi, Yi since hT (xi, x~) : hT (Yl, Yi) ~ 0 implies that hT =-- 0 on M~, contrary to choice o fp i . (Remember the convexity of D means (~* d e <: 0 !) Then near Bd (D) (~ N we have from (2) that K'(x i , y~) ~ O. Q.E.D. to Lemma 2.

Now in U1, g(t)----go �9 tea de o ~. An argument like tha t of Lemma 2 again using the formula (*) to obtain formulas similar to (1) and (2) shows

Lemma 3. For g ( t ) ~ go �9 te3de o ~ to have K ' ~ 0 on some neighborhood o/ Bd (D) c~ N, ~ must be a tangential Kil l ing t-~otto in some neighborhood U2 : U1 o/ Bd (D) c~ N.

Using Lemmas 2 and 3 and formula (*) we can now prove Theorem 1. I f g(t) is a local convex deformation with support in D positive at first order, we may assume g(t) has the form g(t) ~ go �9 re3 de o ~ on some neighborhood U2 of Bd(D) c~ N where ~ is a tangential Killing 1-form on U2. Then for any g0-orthonormal pair of "tangential vectors" (x, y) in U2, by formula (*)

e ~ K'(x, y) -~ ~ - [8* (de)(x, x) 8" ~(y, y) -[-

~- 5* (d~o) (y, y) 5* ~ (x, x) - - 2 5* (de) (x, y ) 5*~(x, y)] -- 0

since ~*~ vanishes identically on "tangential vectors" in U2 so Theorem 1 is proven. I t is not at all dear tha t tangential Killing 1-forms exist near Bd (D) (~ _N. I f there

are no tangential Killing 1-forms near Bd (D) n N, then K ' does not have a definite sign (i.e., ~ 0 or <: 0) in Bd (D) c~ N for any local convex deformation with support in D.

A c k n o w l e d g e m e n t . I would like to thank Jean-Pierre Bourguignon for sharing his ideas on metric deformations stemming from [4] and [5J.

References

[1] T. AuxIN, M6triques rieraannienneset courbure.J. Differential Geometry 4, 383--424 (1970). [2] M. B]~RGE~, Trois remarques sur les vari~t6s s courbure sectionelle positive. C. R. Acad. Sci.

Paris, S~r. A 263, 76--78 (1966).

Vol. XXVI, 19g5 Local Convex Deformations 435

[3] S~minaire B~.RGE~, 1970--71, Vari~t~ "s courbure negative. University of Paris 7, notes prepared by Centre de Math~matiques, ~cole Polytechnique, p. 31.

[4] J.-P. Bov~GviG~o~, A. DESCHA~PS et P. SE~TE~AC, Conjecture de H. Hopf sur les produits de vari~t~s. Ann. Sci. ]~cole Norm. Sup. 4 e S~r. 5, 277--302 (1972).

[5] &-P. BOVRGVm~O~, A. D ~ . s c ~ s et P. S~.~TE~AC, Quelques variations particuli~res de m@triques. To appear in Ann. Sci. ]~cole Norm. Sup.

[6] P. E. EH~LICH, Metric deformations of curvature I: local convex deformations. To appear in Geometriae Dedicata.

[7] D. GRO~IOLT. and W. M~.YwR, An exotic sphere with non negative curvature. Ann. of Math., II. Set. 10O. 401--406 (1974).

Eingegangen am 4. 9. 1974

Anschrift des Autors:

Paul E. Ehrlich Department of Mathematics S.U.N.Y. at Stony Brook Stony Brook, New York 11794, USA and Mathematisches Institut der Universits (current address) D-53 Bonn Wegelerstrage 10 Bundesrepublik Deutschland

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