Ricci Flow Does Not Necessarily Preserve Positive Radial Sectional Curvature

We exhibit a one-parameter family of smooth Riemannian metrics on the four-dimensional sphere with strictly positive radial sectional curvature that loses this property when evolved through the Ricci flow. In other words, while the radial sectional curvature of the four-dimensional sphere with any metric from our one-parameter family is strictly positive at initial time, there exists a tangent plane of the sphere such that the radial sectional curvature of that tangent plane is negative some time after when the metric is evolved through the Ricci flow.For our approach, we initially construct a piecewise-smooth metric that has a nonnegative sectional curvature with a strictly negative temporal derivative of sectional curvature for some tangent plane at initial time. Then we will apply gluing, convolutions, and mollifications in order to obtain a smooth approximation of our piecewise-smooth metric, which is still a nonnegative radial sectional curvature of one tangent plane becoming negative when evolved through the Ricci flow. Furthermore, we will deform the metric slightly in such a way that the nonnegative radial sectional curvature at initial time becomes positive, while one of them still becomes negative when evolved through the Ricci flow. From this, we will extract a one-parameter family of metrics that retain these properties