Helical minimal immersions of compact Riemannian manifolds into a unit sphere

Abstract

An isometric immersion of a Riemannian manifold M into a Riemannian manifold M is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose M is a compact Riemannian manifold which admits a minimal helical immersion of order 4 into the unit sphere. If the Weinstein integer of M equals that of one of the projective spaces, then M is isometric to that projective space with its canonical metric. 0. Introduction. In [2], Besse constructed a minimal immersion with a nice property of a strongly harmonic manifold into a sphere. This nice property is that the images of geodesics of the strongly harmonic manifold are of constant curvatures as curves in the sphere, and the curvatures and the osculating orders are independent of geodesics. Immersions with such a property are said to be helical (cf. [8]). Sakamoto [8] studied a helical immersion of a Riemannian manifold M into a unit sphere. The result is that if M is compact, then it is a Blaschke manifold and, moreover, if the helical immersion is minimal, then M is a globally harmonic manifold. It is well known that if a globally harmonic manifold is compact and simply connected, then it is a strongly harmonic manifold (Michel's theorem, cf. [2]). Thus, we can declare that the theory of helical minimal immersions of compact simply connected Riemannian manifolds into a unit sphere is a submanifold version of strongly harmonic manifold theory. Let f: M S(1) be a helical immersion of a compact Riemannian manifold M into a unit sphere S(1). Since M is a Blaschke manifold, all geodesics of M are simply closed and of the same length. Furthermore, if we denote the cut-locus of x E M by Cut(x), then the unit tangent vectors at x of geodesics emanating from x and entering to y E Cut(x) compose a great sphere in the unit tangent sphere at x. The dimension of the great sphere is independent of x and it is equal to 0, 1, 3, 7, or n 1 (n = dim M), which is the index of the first conjugate point y of x (cf. [2], Proposition 5.39 and Theorem 7.23). Geodesics are also helical in the Euclidean Received by the editors March 15, 1984 and, in revised form, June 20, 1984. 1980 Mathematics Subject Classification. Primary 53C40; Secondary 53C42.