Real hypersurfaces in quaternionic projective space

Summary This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.     

复射影空间中迷向Kaehler子流形

讨论了复射影空间中迷向Kaehler流形,运用活动标架法获得关于截面曲率,Ricci曲率和第二基本形式模长的Pinching定理,将相关结果作了一定的推广.     

Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.     

Lagrangian submanifolds in complex projective space CP n

We first prove a basic theorem with respect to the moving frame along a Lagrangian immersion into the complex projective space CP ⁿ . Applying this theorem, we study the rigidity problem of Lagrangian submanifolds in CP ⁿ .     

On real hypersurfaces with harmonic curvature of a quaternionic projective space

We classify certain real hypersurfaces of a quaternionic projective space satisfying some conditions on their Ricci tensors.Research partially supported by DGICYT Grant PS87-0115-C03-02     

关于四元射影空间中的全实2-调和子流形

利用了活动标架法对四元射影空间QPnc中全实2-调和子流形进行了研究,获得了这类子流形成为全实极小子流形的刚性定理,推广了相关文献中的积分不等式．     

Some remarks on pseudo-umbilical totally real submanifolds in a complex projective space

This paper studies the relationship between the pseudo-umbilical totally real submanifolds and the minimal totally real submanifolds in a complex projective space. Two theo- rems which claim that some types of pseudo-umbilical totally real submanifolds must be minimal submanifolds are proved.     

纤维丛的全测地子流形

本文证明了底空间Ｍ是纤维丛Ｐ的全测地子流形；并且在ｄｉｍＰ－ｄｉｍＭ＝２时证明了若Ｐ是平坦的，则Ｐ的每一纤维也是全测地子流形．     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space. Research partially supported by GNSAGA (INdAM) and MIUR of Italy.     

Totally geodesic submanifolds in Lie groups

Summary We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G_F(n,N) in the Lie group G_F(N) defined naturally makes G_F(n,N) a totally geodesic submanifold; (2) The imbedding S⁷→SO(8) defined by octonians makes S⁷a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G_F(N) in the sphere S^{cN^2-1}(√N) of gl(N,F)is minimal. Therefore the natural imbedding G_F(N)<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>→gl(N,F)is formed by the eigenfunctions of the Laplacian on G_F(N).     

Reduction theorem of the codimension of minimal generic submanifolds in a complex projective space

We study compact minimal generic submanifolds of a complex projective space with flat normal connection and prove a reduction theorem of codimension under the condition on the Ricci tensor.The present studies were supported by the Basic Science Research Institute Program, Korea Ministry of Education, 1993-114.