Warping functions of some warped products

In this paper we study the warping functions of some warped products. Let HnHn be the n-dimensional hyperbolic space with sectional curvature −1. We prove that if the warping function f   of the warped product Hn×fRHn×fR has a critical point, then Hn×fR=Hn+1Hn×fR=Hn+1 if and only if f(x)=kcoshr(x), where k   is a positive constant, r(x)r(x) denotes the hyperbolic distance from x   to a fixed point. We also prove that if the sectional curvature of the warped product M×fRM×fR is nonnegative, then f is constant, providing that the Riemannian manifold M is complete and connected.