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Positively curved deformations of invariant Riemannian metrics

Let K, denote the sectional curvature function of the Riemannian metric y on a manifold M. Suppose M admits no metric -y invariant under the action of a compact group G and having K, >O. It is shown that a G-invariant metric -y(0) with K,(o)_0 cannot be embedded in a 1-parameter family y(t) for which [dKy(t)/dt]t..o is positive wherever K,(o) is zero. The sectional curvature of the Riemannian metric y on the manifold M is a function K, on the bundle P(M) of tangent 2-planes to M. Denote by Z,CP(M) the set of zeros of K. A smooth 1-parameter family y(t) will be called positive if K,(o) is nonnegative and if [dK_y(t)/dt]tCo is positive on Z,(o). If M is compact, K,(E) is positive for small positive E. Berger [2 ] showed that if M is compact, a family on M for which y(O) is a product metric cannot be positive. The following theorem, combined with the results cited after its proof, gives other examples of metrics which cannot be embedded in positive families. THEOREM. If M is a compact manifold which admits no metric of positive sectional curvature invariant under a given compact group G of diffeomorphisms, then a family y(t) of metrics on M for which y(0) is G-invaritant cannot be positive. PROOF. We prove the contrapositive. Suppose that y(t) is G-invariant, and define