Submanifolds of the Cayley projective plane with bounded second fundamental form

Abstract

We consider totally complex submanifolds of the Cayley projective plane with estimates on the length squared of the second fundamental form. We determine those bounds for which the second fundamental form is parallel and for which the submanifold is totally geodesic. The case of totally real submanifolds is also included. O. INTRODUCTION Let M be a smooth m-dimensional Riemannian manifold isometrically immersed in an (m + p)-dimensional Riemannian manifold N. Let h denote the second fundamental form of this immersion. For each x ~ M, h is a bilinear mapping from TxM x TxM into TxM l, where T~M is the tangent space of M, and TxM ± is the normal space. We denote by S(x) the length squared of h at x E M. The Gauss equation shows that S(x) = m(m -1) -fl(x), where fl(x) is the scalar curvature of M at x. In this way S(x) is an intrinsic invariant of M. In [13] Simons showed that if M is minimal in S re+p, the totally geodesic submanifolds are isolated in so far that if S(x) ~< m/(2 -1/p) for all x ~ M, then S(x) = 0 identically on M, and M is totally geodesic. All minimal submanifolds of S "+p satisfying S(x) = m/(2 -l/p) were determined in [7]. Subsequently, analogous results for various minimal submanifolds of the complex and quaternionic projective space were obtained.Consider now the unit tangent bundle UM. We set 6(u) = IIh(u, u)l12 for u~ UM. Unlike S(x), 6(u) is not an intrinsic invariant of M, but can be considered as a natural measure of the degree to which an immersion fails to be totally geodesic.For CP n, Ros [11] showed that ifM is a compact Kaehler submanifold and if 6(u) < ¼, for any u ~ UM, then M is totally geodesic. He also lists those Kaehler submanifolds satisfying the condition maxu~vu{f(u)} = ¼. Coulton and Gauchman proved the same estimate for totally complex submanifolds of HP n, the quaternionic projective space [8]. Recall that totally complex submanifolds of HP ~ inherit a KaeMer structure in a natural way. Tsukada [14] has given a classification of parallel submanifolds of HP" which satisfy 6(u) = ¼. Analogous results have also been obtained [8] in the case of totally real immersions into HP n.There remains the rank 1 compact irreducible symmetric space, the Cayley Geometriae Dedicata 33: [265][266][267][268][269][270][271][272][273][274][275] 1990.

Totally complex submanifolds of the Cayley projective plane

“…Moreover in [6], Ros gave a complete list of compact Kaehler submanifolds of C7""(l) satisfying the condition l^M A u ) = \-The same type results for totally complex submanifolds of the quaternion projective space HP"\\) were obtained by Coulton and Gauchman [3]. In [4], Coulton and Glazebrook proved the analogous results in the case of totally complex submanifolds of the Cayley projective place CaP 2 . In the present paper, we proved the following pinching theorem for the square of the norm of the second fundamental form.…”

“…In this section, we review the fundamental results about the Cayley projective plane; for details see [4].…”