On the Structure of the Cut Locus in Certain Classes of Warped Product Manifolds

In this paper, we determine the structure of the cut locus for a class of complete warped product manifolds $M=(R^1\times S^1,d{t}^2+m(t{)}^{2}d{\theta }^2)$, where the warping function $m(t)$ is positive and even, under the assumption that there exists a positive number $t_1$ such that the Gaussian curvature is nonpositive on the domain $t^{-1}((t_1,\mathrm{\infty }))$ and is decreasing along the subarc $t^{-1}([0,t_1])\cap {\theta }^{-1}(0)$ of the upper half meridian.