Planar geodesic submanifolds in a quaternionic projective space

The planar geodesic submanifolds of a quaternionic projective space are studied. Especially, these submanifolds which are totally real or quaternionic CR-submanifolds are completely classified. Also, the non-existence of a planar geodesic, proper QR-product in a quaternionic projective space is proved.Research supported in part by a grant from KOSEF.     

Hyper-Hermitian Quaternionic Kähler Manifolds

We call a quaternionic Kähler manifold with nonzero scalar curvature, whosequaternionic structure is trivialized by a hypercomplex structure, ahyper-Hermitian quaternionic Kähler manifold. We prove that every locallysymmetric hyper-Hermitian quaternionic Kähler manifold is locally isometricto the quaternionic projective space or to the quaternionic hyperbolic space.We describe locally the hyper-Hermitian quaternionic Kähler manifolds withclosed Lee form and show that the only complete simply connected suchmanifold is the quaternionic hyperbolic space.     

关于四元射影空间的正曲率全复子流形

本文讨论四元射影空间的全复子流形，证明了四元射影空间的正截面曲率紧致全复子流形一定是全测地的。     

Envelopes and osculates of Willmore surfaces

We view conformal surfaces in the 4-sphere as quaternionic holomorphiccurves in quaternionic projective space. By constructing envelopingand osculating curves, we obtain new holomorphic curves in quaternionicprojective space and thus new conformal surfaces. Applying theseconstructions to Willmore surfaces, we show that the osculatingand enveloping curves of Willmore spheres remain Willmore.     

Eigenvalue estimates for the Dirac operator on quaternionic K?hler manifolds

We consider the Dirac operator on compact quaternionic K?hler manifolds and prove a lower bound for the spectrum. This estimate is sharp since it is the first eigenvalue of the Dirac operator on the quaternionic projective space. Received April 21, 1998; in final form June 16, 1998     

Classification of conformal minimal immersions of constant curvature from S^2 to { HP}^2

In this paper, we study geometry of conformal minimal two-spheres immersed in quaternionic projective spaces. We firstly use Bahy-El-Dien and Wood’s results to obtain some characterizations of the harmonic sequences generated by conformal minimal immersions from \(S^2\) to the quaternionic projective space \({ HP}^2\) . Then we give a classification theorem of linearly full totally unramified conformal minimal immersions of constant curvature from \(S^2\) to the quaternionic projective space \({ HP}^2\) .     

Harmonic tori in quaternionic projective 3-spaces

Burstall classified conformal non-superminimal harmonic two-tori in spheres and complex projective spaces. In this paper, we shall classify conformal non-superminimal harmonic two-tori in a 2- or 3-dimensional quaternionic projective space, which are not always covered by primitive harmonic two-tori of finite type.

     

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     

关于四元射影空间的一般极小子流形

本文研究四元射影空间的一般极小子流形,建立了一个积分公式,并且用此公式得到了紧致一般极小子流形的截曲率的Ｐｉｎｃｈｉｎｇ结果．     

Scalar curvature of <Emphasis Type="Italic">QR</Emphasis>-submanifolds with maximal <Emphasis Type="Italic">QR</Emphasis>-dimension in a quaternionic projective space

In this paper we derive an integral formula on an n-dimensional, compact, minimal QR-submanifoldM of (p−1) QR-dimension immersed in a quaternionic projective space QP ^(n+p)/4. Using this integral formula, we give a sufficient condition concerning with the scalar curvature of M in order that such a submanifold M is to be a tube over a quaternionic projective space.     

Smooth Involutions on the 2k-dimensional Quaternionic Projective Space HP(2k)

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Twistor theory and Blaschke manifolds

Using the twistor theory on quaternionic Kaehler manifolds and some recent results on Blaschke manifolds and compact manifolds whose holonomy group is Spin (7), we prove that a Blaschke manifold of nonnegative scalar curvature whose holonomy group is exceptional is isometric to a projective space.     

Diffeomorphisms taking lines to circles, and quaternionic Hopf fibrations

We list all diffeomorphisms between an open subset of the four-dimensional projective space and an open subset of the four-dimensional sphere that take all line segments to arcs of round circles. These diffeomorphisms are restrictions of quaternionic Hopf fibrations and radial projections from hyperplanes to spheres.     

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.     

Minimal two-spheres with constant curvature in the quaternionic projective space

In this paper we completely classify the homogeneous two-spheres, especially, the minimal homogeneous ones in the quaternionic projective space HPn. According to our classification, more minimal constant curved two-spheres in HPnare obtained than what Ohnita conjectured in the paper "Homogeneous harmonic maps into complex projective spaces. Tokyo J Math, 1990, 13: 87–116".     

Scalar-flat K?hler orbifolds via quaternionic-complex reduction

We prove that any asymptotically locally Euclidean scalar-flat K?hler 4-orbifold whose isometry group contains a 2-torus is isometric, up to an orbifold covering, to a quaternionic-complex quotient of a k-dimensional quaternionic vector space by a (k−1)-torus. In order to do so, we first prove that any compact anti-self-dual 4-orbifold with positive Euler characteristic whose isometry group contains a 2-torus is conformally equivalent, up to an orbifold covering, to a quaternionic quotient of k-dimensional quaternionic projective space by a (k − 1)-torus.     

QR-Submanifolds of (p - 1) QR-Dimension in a Quaternionic Projective Space QP (n+p)/4

The purpose of this paper is to study n-dimensional QR-submanifolds of (p - 1) QR-dimension isometrically immersed in a quaternionic projective space QP ^(n+p)/4 and to give sufficient conditions in order for such a submanifold to be a tube over a quaternionic invariant submanifold.     

The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.     

Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

We consider the mean curvature flow of a closed hypersurface in the complex or quaternionic projective space. Under a suitable pinching assumption on the initial data, we prove apriori estimates on the principal curvatures which imply that the asymptotic profile near a singularity is either strictly convex or cylindrical. This result generalizes to a large class of symmetric ambient spaces the estimates obtained in the previous works on the mean curvature flow of hypersurfaces in Euclidean space and in the sphere.     

Quaternionic Killing Spinors

In [17] we proved a lower bound for the spectrum of the Dirac operator on quaternionic Kähler manifolds. In the present article we study the limiting case, i.e. manifolds where the lower bound is attained as an eigenvalue. We give an equivalent formulation in terms of a quaternionic Killing equation and show that the only symmetric quaternionic Kähler manifolds with smallest possible eigenvalue are the quaternionic projective spaces.