Constant isotropic submanifolds with 4-planar geodesics

Abstract

ABSTRACT. Let / be an isometric immersion of a Riemannian manifold M into M. We prove that if / is constant isotropic, 4-planar geodesic and M is a Euclidean sphere, then M is isometric to one of compact symmetric spaces of rank equal to one and / is congruent to a direct sum of standard minimal immersions. We also determine constant isotropic, 4-planar geodesic, totally real immersions into a complex projective space of constant holomorphic sectional curvature. Introduction.In [15], O'Neill studied isotropic immersions and showed that if the difference between the sectional curvatures of the submanifold and the ambient manifold is constant for any plane section tangent to the submanifold, then the codimension must be high compared with the dimension of the submanifold unless the immersion is totally geodesic or umbilical. The class of (constant) isotropic immersions (for instance, into a sphere) seems to be too large to classify them. For example, the composition of two isotropic immersions is also isotropic and for two given isotropic immersions fy and f2 into 5P(1) and 5a(l), respectively, their "direct sum" (cos8fy,sin8f2) into 5p+,+1(l) is also isotropic for any 9. Furthermore, it is well known that there are many constant isotropic immersions in equivariant isometric immersions of homogeneous spaces.An isometric immersion /: A_ -► M is said to be 4-planar geodesic if, for each geodesic r of M, the curve / o 7 is locally contained in a 4-dimensional totally geodesic submanifold of M. The class of 4-planar geodesic immersions (into a sphere) is also large. At least, an isometric immersion of a surface into a 4-dimensional Riemannian manifold is always 4-planar geodesic.In this paper, we try to classify immersions in the intersection of these classes. In particular, we are able to classify constant isotropic, 4-planar geodesic (resp. and totally real) immersions of connected, complete Riemannian manifolds into a unit sphere (resp. complex projective space of constant holomorphic sectional curvature 4).In §1, we give basic equations used later and some definitions. In §2, we prove that if / is a constant isotropic, 4-planar geodesic (resp. and totally real) immersion into a real space form (resp. CPq(c)), then / is a helical immersion of order < 4. Here we recall that an isometric immersion /: M -» M is called a helical immersion