A Generalization of Minimal Cones

Let R+ be a positive real line, S" an -dimensional unit sphere.We denote by R + XS" the polar coordinate of an (n + l)-dimensional Euclidean space R" + 1.It is well known that if M is a minimal submanifold in S", then R + XM is minimal in Rn+.R + XM is called a minimal cone.We generalize this fact and give many minimal submanifolds in real and complex space forms.1. Introduction.The existence and global behavior of submanifold with constant mean curvature is, in full generality, a difficult area of study.The nonlinearity of the problem makes even the construction of explicit examples reasonably difficult and, at the same time, makes such examples indispensable guidelines for research.Let M be a Riemannian manifold and G a compact connected group of isometries of M. Then W. Y. Hsiang and H. B. Lawson, Jr. [7 proved that a G-invariant submanifold N of M is minimal if and only if N/G is minimal in M/G with the appropriate metric.Under this observation, they explicitly construct vast numbers of compact minimal submanifolds in nearly all homogeneous spaces.In 13, J. Simons gives a powerful method of analyzing compact minimal submanifolds in a sphere and a complex projective space.Nowadays it is known that there exist many pinching theorems.It is natural that we are interested in the relaxation among those pinching's values.The purpose of this paper is to give two observations for the construction of complete submanifolds with constant mean curvature in a real and complex space form and to see the relation between examples obtained by our discussion and some pinching theorems.Our two observations are as follows: First observation.Let B and F be, respectively, a Riemannian manifold with metric g, and a Riemannian manifold with metric g2.Let/be a positive function on B. We denote by B XfF the product manifold B X F of B and F with metric g = gx + f2g2.B X, F is called the warped product of B X F 1.Let M and N be an w-dimensional submanifold of B and an -dimensional submanifold of F, respectively.We may naturally see that M X N is an (i + )-dimensional submanifold of B XfF, and, with respect to the induced metric, it is also a warped product M XfN.