. In this paper, we show new results on slant submanifolds of analmost contact metric manifold. We study and characterize slant submanifolds of K-contact and Sasakian manifolds. We also study the special class of three-dimen-sional slant submanifolds. We give several examples of slant submanifolds.1991 Mathematics Subject Classiﬁcation. 53C15, 53C40.0. Introduction. Slant immersions in complex geometry were deﬁned by B.-Y.Chen as a natural generalization of both holomorphic immersions and totally realimmersions [2]. Examples of slant immersions into complex Euclidean spaces C 2 andC 4 were given by Chen and Tazawa [2, 4, 5], while slant immersions of Ka¨hler C-spaces into complex projective spaces were given by Maeda, Ohnita and Udagawa[9].In a recent paper [7], A. Lotta has introduced the notion of slant immersion of aRiemannian manifold into an almost contact metric manifold and he has provedsome properties of such immersions. A. Lotta and A. M. Pastore have obtainedexamples of slant submanifolds in the Sasakian-space-form R

/pdf/slant-submanifolds-in-sasakian-manifolds-1ead3jmch5.pdf

Slant submanifolds in sasakian manifolds

The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.

Real Hypersurfaces in Complex Space Forms

Manifolds all of whose geodesics are closed

We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold $${\bar{M}}$$
 in $${\mathbb{C}P^n}$$
 and the bitension field of the inclusion of the corresponding Hopf-tube in $${\mathbb{S}^{2n+1}}$$
 . Using this relation we produce new families of proper-biharmonic submanifolds of $${\mathbb{C}P^n}$$
 . We study the geometry of biharmonic curves of $${\mathbb{C}P^n}$$
 and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.

Biharmonic submanifolds of ${\mathbb{C}P^n}$

Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given

On Ruled Real Hypersurfaces in Complex Space Forms

This paper consists of two parts. One is to construct a class of helical geodesic equivariant immersions of orderd(⩾3), which are neither Kaehler nor totally real immersions, into complex projective spaces. The other is to show the basic results about a helix in complex space forms.

Helical geodesic immersions into complex space forms

for some positive constant k and a field of unit vectors Ys along y. Here Vs denotes the covariant differentiation along y with respect to the Riemannian connection V of M. The positive constant k is called the curvature of y. For given a positive k and an orthonormal pair of vectors u, v e TXM at a given point x e M, we have a unique circle y defined for ―oo ) we have an

/pdf/global-behaviours-of-circles-in-a-complex-hyperbolic-space-3na9m8uuou.pdf

Global behaviours of circles in a complex hyperbolic space

/pdf/length-spectrum-of-geodesic-spheres-in-a-non-flat-complex-1y8f8qpm30.pdf

Length spectrum of geodesic spheres in a non-flat complex space form

In this paper we study congruency of minimal ruled real hypersurfaces in a nonflat complex space form with respect to the action of its isometry group. We show that those in a complex hyperbolic space are classified into 3 classes and show that those in a complex projective space are congruent to each other hence form just one class.

/pdf/congruence-classes-of-minimal-ruled-real-hypersurfaces-in-a-3xfvueoraa.pdf

Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form

Let φ: M-*N be an isometric immersion of a Riemannian manifold into an almost Hermitian manifold with almost complex structure /. Then, φ is called slant if the angle between jφ*(X) and <p*(TpM) is constant for any X^TPM and any p^M. The typical examples of slant immersions are Kahler immersions and totally real immersions, where the slant angles are 0 and π/2, respectively. A slant immersion is called proper if it is neither a Kahler immersion nor a totally real immersion. In the case where M is a Riemann surface and N is a Kahler manifold, the slant angle was introduced as the Kdhler angle and studied by S. S. Chern and J. G. Wolf son [CW]. Examples of slant immersions of a Riemann sphere S into a complex projective space CP were given as the Veronese sequence of harmonic maps from S, which are classified in [BO] and [BJRW] in the case where S has constant curvature (see also [01]). The present concept of slant immersion was first introduced and studied by B. Y. Chen [C]. The examples of proper slant immersions into C are given in [C-T], Recently, Tazawa [T] has given examples of slant immersions into C with any given slant angle. However, there are a few results on slant submanifolds in CP. In this case, any general method to check whether given an immersion is slant or not is not known.

https://www.med.nihon-u.ac.jp/home/sudagawa/MOU-Slant.pdf

Sadahiro Maeda

Papers

Helical geodesic immersions into complex space forms

Global behaviours of circles in a complex hyperbolic space

Length spectrum of geodesic spheres in a non-flat complex space form

Congruence classes of minimal ruled real hypersurfaces in a nonflat complex space form

On slant immersions into Kähler manifolds