PAUL EHRLICH*

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

The point of this note, an appendix to [2], is to remark that the proof of the Ricci curvature deformation theorem by the method of local convex deformations, Theorem 1, Section 5 of [2], yields without any modification.

T H E O R E M 1. Let M", n >~ 2, be a smooth manifold. Suppose M admits a eomplete C r Riemannian metric, 4<~r<<.oo, with nonnegative (nonpositive) scalar curvature and some point of positive (negative) scalar curvature. Then

M admits a complete C ~ Riemannian metric of everywhere positive (negative)

scalar curvature. Remark. Even for M compact, Theorem 1 seems to have been overlooked

in the case of nonnegative scalar curvature. We note that in the compact case, Theorem 1 for nonpositive scalar

curvature is implied by the stronger

T H E O R E M 2 (Yamabe, Aubin, Elliason, see [3]). Let M", n>>.3, be a com- pact smooth manifold. Then M adm#s a smooth Riemannian metric w#h con- stant negative scalar curvature. and also from Theorem 6.4 of [4] which asserts that if d i m M > 3, any smooth function negative somewhere on M is the scalar curvature function of some smooth Riemannian structure. Theorem 1 in the compact case for d i m M = 2 follows from Theorem 5.1 of [4].

For dimM~>3, our Theorem 1 for nonnegative scalar curvature in the compact case follows from

T H E O R E M 3 (Bourguignon, Fischer, Marsden, see [1]). Let M", n>_ 3, be a

smooth compact manifold. Let R(M) denote the space of smooth Riemannian metric for M. Then the scalar curvature mapping

: R ( M ) --, C ~ ( M )

is an I.L.H. submersion except at metrics g for which Ric(g)_0 or for which T(g)--- constant > 0. Frank Warner has also remarked (personal communication) that our Theorem 1 in the compact case follows also from the methods of [4], [5].

* This work was done under the program 'SonderforschungsbereichTheoretische Mathe- matik' (SFB 40) at the University of Bonn.

Geometriae Dedicata 5 (1976) 25-26. All Rights Reserved Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland

26 PAUL EHRLICH

As an application of Theorem 1, we give a constructive proof of Theorem 4 below. This result also follows non-constructively from the submersion theorem for the scalar curvature mapping of Bourguignon, Fischer, and Marsden stated above.

THEOR EM 4. Let M', n t> 3, be a compact smooth manifoM admitting a

Riemannian metric g of nonnegative scalar curvature. Suppose at some point

where the scalar curvature o f g is zero, some Ricci curvature o f g is non-zero.

Then M admits a metric o f everywhere positive scalar curvature.

Proof First by a result of Bourguignon, [1], part b, Section 8, Propo- sition 8.4 and Remark 8.6, our assumptions on the Ricci tensor and scalar curvature imply that the deformation g(t): = g - t Ric(g) produces a Rieman- nian metric for M with nonnegative scalar curvature and with a point o f positive scalar curvature. Then apply Theorem 1. Q.E.D.

This proof of Theorem 4 also gives a proof of the following result which is really just a restatement of Theorem 4. The result was noticed by Bourguignon (unpublished) and proved by a different method by Kazdan and Warner in [5], Lemma 5.2, using a differential operator suggested by the curvature formulas for conformal changes of metrics.

THEOREM 5. Suppose M ' , n~>3, compact, admits no metric o f positive

scalar curvature. Then any metric for M with zero scalar curvature must be

Ricei flat.

BIBLIOGRAPHY

1. Bourguignon, Thrse, Universit6 de Paris 7, L'espace des structures riemanniennes d'une varidtd eompacte, 27 mai, 1974, also to appear in Composito Math.

2. Ehrlich, 'Metric Deformations of Curvature. I: Local Convex Deformations', this issue of Geometriae Dedieata.

3. Elliason, 'On Variations of Metrics', Math. Scand. 29 (1971), 317-327. 4. Kazdan and Warner, 'Existence and Conformal Deformation of Metrics with Prescribed

Gaussian and Scalar Curvature', to appear in the Annals of Math. 5. Kazdan and Warner, 'Prescribing Curvature', to appear in the Proceedings of the Amer.

Math. Soe. Summer Institute in Differential Geometry, Stanford, 1973.

Author's address:

Paul E.Ehrlich, Mathematisches Institut der Universit~it Bonn 53 Bonn, WegelerstraBe 10, Federal Republic of Germany

(Received February 1, 1975)