Lengths of Closed Geodesics for Certain Warped Product Metrics and Nearly Round Metrics on Spheres

We study warped product metrics on spheres and prove that the shortest length of closed geodesics for such metric can be bounded above by its volume under suitable assumptions on the warping function. As a result we manage to extend the classical inequality on two-spheres between the shortest closed geodesic and the area to certain higher dimensional spheres. In particular we establish such systolic inequalities for smooth closed rotational hypersurfaces in Euclidean spaces or unit spheres. Moreover, we also establish similar inequalities for metrics on higher-dimensional spheres that are close to the round metric.