Submanifolds of quaternion projective space with bounded second fundamental form

Abstract

Let h be the second fundamental form of a compact submanifold M of the quaternion projective space HP n (Y). For any unit vector u^TM, set δ(u) = \\h(u,u)\\ 2 . We determine all compact totally complex submanifolds of HP 71 (I) (resp. all compact totally real minimal submanifolds of HP n (l)) satisfying condition δ(u)^-j-(resp. δ(u)^-)for all unit vectors 1. Introduction.Let M be a smooth m-dimensional Riemannian manifold isometrically immersed in an (m+jfr)-dimensional Riemannian manifold M. Let h denote the second fundamental form of this immersion. For each IGM, /ι is a bilinear mapping from TM x xTM x into TMi, where TM X is the tangent space of M at x and TMi is the normal space. We denote by S(x) the square of the length of h at IGM. By Gauss' equation we have S(x)=m(m-1)-p(x), whenever M is immersed as a minimal submanifold of S m+P (l) with scalar curvature p(x) at x in M. Therefore S(x) is an intrinsic invariant of M.In 1968, J. Simons [12] discovered for the class of compact minimal m-dimensional submanifolds of the unit (m+£)-sphere that the totally geodesic submanifolds are isolated in the following sense: If S(x)<n/(2-l/p) for all XGM, then S(x)=0 on M, and thus M is totally geodesic. In [1], S. S. Chern, M. do Carmo, and S. Kobayashi determined all minimal submanifolds of the unit sphere satisfying S(x)=n/(2-l/p).Later similar results were obtained for various types of minimal submanifolds of the complex projective spaces and the quaternion projective spaces.Let T: UM-+M and UM X denote the unit tangent bundle of M along with its fibre over xεM We set δ{u)=\\h(u, u)\\ 2 for u^UM:Observe that δ(u) is not an intrinsic invariant of the submanifold M. However, like S(x), δ(u) can be considered as a natural measure of the degree to which an immersion fails to be totally geodesic.In [10], and [11], A. Ros proved that if M is a compact Kaehler submanifold of CP n (l) and if δ(w)<l/4, for any u^UM, then M is totally geodesic in

Totally complex submanifolds of the Cayley projective plane

“…Denote by It the second fundamental form of M and UM the unit tangent bundle over M. Ros showed in [5] that \ff[u) = \h(u, w)| 2 < | for any u e UM, then M is totally geodesic. 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 .…”

Totally complex submanifolds inH P m (1)

“…Moreover in [6], Ros gave a 4 complete list of compact Kaehler submanifolds of CP m (1) satisfying the condition maxu~uM ~5(u) = 41-. Coulton and Gauchman in [1] proved the same type results for totally complex submanifolds of HPm(1), the quaternion projective space. In the present paper, we prove some pinching theorems for the square norm of the second fundamental form and the sectional curvature of compact totally complex submanifolds in H P m (1).…”

“…Let N be a 4m-dimensional differentiable manifold, and assume that there is a 3-dimensional vector bundle V, [4], consisting of tensors of type (1,1) Assume that the Riemannian connection V of(N, V, g) satisfies the following condition: if ¢ is a local cross-section of the bundle V, then Vx¢ is also a local cross-section of V, where X is an arbitrary vector field. In this case N = (N, V, g) is called a Kaehler quaternion manifold.…”

“…Let M be an n-dimensional compact Kaehler submanifold immersed in a complex projective space CP ~ (1). Denote by cr the second fundamental form of M and UM the unit tangent bundle over M. Ros in [5] showed that if ~5(u) := [(r(u, u)] 2 < !…”

“…7-: cpm(1)--+ Hpm (1), along with the following standard embeddings [9]: ~l,n: n 1 CP (g) ~ CPk (1), where k = n(n+3)/2 ~2: Qn ~ Cpn+l(1), n > 3 and Q is the standard quadric.…”