Positively Curved Deformations of Invariant Riemannian Metrics

Abstract

Let ${K_\gamma }$ denote the sectional curvature function of the Riemannian metric $\gamma$ on a manifold $M$. Suppose $M$ admits no metric $\gamma$ invariant under the action of a compact group $G$ and having ${K_\gamma } > 0$. It is shown that a $G$-invariant metric $\gamma (0)$ with ${K_{\gamma (0)}} \geqq 0$ cannot be embedded in a $1$-parameter family $\gamma (t)$ for which ${[d{K_{\gamma (t)}}/dt]_{t = 0}}$ is positive wherever ${K_{\gamma (0)}}$ is zero.