Warping functions of some warped products

In this paper we study the warping functions of some warped products. Let H n be the n-dimensional hyperbolic space with sectional curvature −1. We prove that if the warping function f of the warped product H n × f R has a critical point, then H n × f R = H n + 1 if and only if f ( x ) = k cosh r ( x ) , where k is a positive constant, 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 × f R is nonnegative, then f is constant, providing that the Riemannian manifold M is complete and connected.