HARMONIC FOLIATIONS ON A COMPACT RIEMANNIAN MANIFOLD OF NON-NEGATIVE CONSTANT CURVATURE

Introduction. Let M be a compact oriented manifold and ^~ a Riemannian and harmonic foliation with respect to a bundle-like metric. Kamber and Tondeur [3] proved the fundamental formula for a special variation of ^ and making use of it they showed in [4] that the index of a Riemannian and harmonic foliation on the sphere S (n > 2) for which the standard metric is bundle-like is not smaller than q + 1, where q is the codimension of j^~. The purpose of this paper is to prove that any harmonic foliation on a compact Riemannian manifold of non-negative constant curvature for which the normal plane field is minimal (see § 1 for the definition) is totally geodesic. As a corollary we can state that any Riemannian and harmonic foliation on the sphere S (n > 2) for which the standard metric is bundlelike is totally geodesic. Moreover, Escobales [1] has classified recently all totally geodesic foliations on the spheres for which the standard metrics are bundle-like. This means that harmonic foliations on the spheres for which the standard metrics are bundle-like have been completely classified. On the other hand, a theorem of Ferus [2] gives an estimate for the codimension of a totally geodesic foliation of the sphere S. Thus we can apply these results to the foregoing theory of Kamber and Tondeur to sharpen their result. The authors wish to thank the referee for his useful advice.