We establish a general sharp inequality for warped products in real space form. As applications, we show that if the warping function $f$ of a warped product $N_1\times_fN_2$ is a harmonic function, then
(1) every isometric minimal immersion of $N_1\times_fN_2$ into a Euclidean space is locally a warped-product immersion, and
(2) there are no isometric minimal immersions from $N_1\times_f N_2$ into hyperbolic spaces.
Moreover, we prove that if either $N_1$ is compact or the warping function $f$ is an eigenfunction of the Laplacian with positive eigenvalue, then $N_1\times_f N_2$ admits no isometric minimal immersion into a Euclidean space or a hyperbolic space for any codimension. We also provide examples to show that our results are sharp.
AMS 2000 Mathematics subject classification: Primary 53C40; 53C42; 53B25
