Quaternionic Space Forms Research Articles

Chen-Type Inequality for Generic Submanifolds of Quaternionic Space Form and Its Application

In 1993, the theory of Chen invariants started when Chen wrote basic inequalities for submanifolds in space forms. This inequality is called Chen’s first inequality. Afterward, many geometers studied many papers dealing with this new inequality. The present paper aims to establish a Chen inequality for quaternionic generic submanifolds in a quaternionic space form and obtain this inequality for real hypersurfaces.

δ(2,2)-Invariant for Lagrangian Submanifolds in Quaternionic Space Forms

In the geometry of submanifolds, Chen inequalities represent one of the most important tool to find relationships between intrinsic and extrinsic invariants; the aim is to find sharp such inequalities. In this paper we establish an optimal inequality for the Chen invariant δ ( 2 , 2 ) on Lagrangian submanifolds in quaternionic space forms, regarded as a problem of constrained maxima.

Generalized Wintgen inequality for Lagrangian submanifolds in quaternionic space forms

Generalized Wintgen inequality for Lagrangian submanifolds in quaternionic space forms

An Inequality on Quaternionic CR-Submanifolds

Abstract We establish an inequality for an intrinsic invariant of Chen-type defined on quaternionic CR-submanifolds in quaternionic space forms, in terms of the squared mean curvature, the main extrinsic invariant, by using the method of constrained extrema.

Harmonicity and minimality of complex and quaternionic radial foliations

Abstract We construct special classes of totally geodesic almost regular foliations, namely, complex radial foliations in Hermitian manifolds and quaternionic radial foliations in quaternionic Kähler manifolds, and we give criteria for their harmonicity and minimality. Then examples of these foliations on complex and quaternionic space forms, which are harmonic and minimal, are presented.

Ricci solitons on QR-hypersurfaces of a quaternionic space form ℚn

Abstract The purpose of this paper is to study Ricci solitons on QR-hypersurfaces M of a quaternionic space form ℚn such that the shape operator A with respect to N has one eigenvalue. We prove that Ricci soliton on QR-hypersurfaces M with eigenvalue zero is steady and for eigenvalue nonzero is shrinking.

A new proof for some optimal inequalities involving generalized normalized δ-Casorati curvatures

In this paper we give a new proof for two sharp inequalities involving generalized normalized δ-Casorati curvatures of a slant submanifold in a quaternionic space form. These inequalities were recently obtained in Lee and Vîlcu (Taiwan. J. Math. 19(3):691-702, 2015) using an optimization procedure by showing that a quadratic polynomial in the components of the second fundamental form is parabolic. The new proof is obtained analyzing a suitable constrained extremum problem on submanifold.

INEQUALITIES FOR GENERALIZED NORMALIZED $\delta$-CASORATI CURVATURES OF SLANT SUBMANIFOLDS IN QUATERNIONIC SPACE FORMS

In this paper we prove two sharp inequalities involving the normalized scalar curvature and the generalized normalized $\delta$-Casorati curvatures for slant submanifolds in quaternionic space forms. We also characterize those submanifolds for which the equality cases hold. These results are a generalization of some recent results concerning the Casorati curvature for a slant submanifold in a quaternionic space form obtained by Slesar et al.: J. Inequal. Appl. 2014, 2014:123.

Remarks on inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms

In this paper, we obtain an inequality for the normalized Casorati curvature of slant submanifolds in quaternionic space forms by using T Oprea’s optimization method.MSC:53C40, 53D12.

Inequalities for the Casorati curvatures of slant submanifolds in quaternionic space forms

Abstract In this paper we prove two sharp inequalities that relate the normalized scalar curvature with the Casorati curvature for a slant submanifold in a quaternionic space form. Moreover, we show that in both cases, the equality at all points characterizes the invariantly quasi-umbilical submanifolds.

CERTAIN CLASS OF QR-SUBMANIFOLDS OF MAXIMAL QR-DIMENSION IN QUATERNIONIC SPACE FORM

In this paper we determine certain class of <TEX>$n$</TEX>-dimensional QR-submanifolds of maximal QR-dimension isometrically immersed in a quaternionic space form, that is, a quaternionic K<TEX>$\ddot{a}$</TEX>hler manifold of constant Q-sectional curvature under the conditions (3.1) concerning with the second fundamental form and the induced almost contact 3-structure.

On Chen invariants and inequalities in quaternionic geometry

In this paper we give a survey of main results concerning Chen inequalities for submanifolds in quaternionic space forms and propose some open problems in the field for further research. AMS Subject Classification:53C15, 53C25, 53C40.

Slant submanifolds of quaternionic space forms

In this paper we establish some inequalities concerning the $k$-Ricci curvature of a slant submanifold in a quaternionic space form. We also obtain obstructions to the existence of quaternionic slant immersions in quaternionic space forms with unfull first normal bundle.

Cellular decomposition of quaternionic spherical space forms

We obtain an explicit cellular decomposition of the quaternionic spherical space forms, manifolds of positive constant curvature that are factors of an odd sphere by a free orthogonal action of a generalized quaternionic group. The cellular structure gives and explicit description of the associated cellular chain complex of modules over the integral group ring of the fundamental group. As an application we compute the Whitehead torsion of these spaces for any representation of the fundamental group.

Curvature-adapted submanifolds of symmetric spaces

Curvature-adapted submanifolds have been extensively studied in complex and quaternionic space forms. This paper extends their study to a wider class of ambient spaces. We generalize Cartan's theorem classifying isoparametric hypersurfaces of spheres to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. We classify those with constant principal curvatures in the octonionic planes. Various classification results for hypersurfaces in complex two-plane Grassmannians are also obtained.

B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms

In this paper some B.-Y. Chen inequalities for slant submanifolds in quaternionic space forms are established.

Ruled Real Hypersurfaces in a Nonflat Quaternionic Space Form

In this paper we construct many ruled real hypersurfaces in a nonflat quaternionic space form systematically, and in particular give an example of a homogeneous ruled real hypersurface in a quaternionic hyperbolic space. In the second half of this paper we characterize them by investigating the extrinsic shape of their geodesics. We also characterize curvature-adapted real hypersurfaces in nonflat quaternionic space forms from the same viewpoint.

Powers of the curvature operator of space forms and geodesics of the tangent bundle

It is well known that if Г is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold, then all geodesic curvatures of the projected curve λ=π 1463-01 Г are constant. In this paper, we consider the case of the tangent (sphere) bundle over real, complex, and quaternionic space forms and give a unified proof of the following property: All geodesic curvatures of the projected curve are zero beginning with k 3, k 6, and k 10 for the real, complex, and quaternionic space forms, respectively.

On the Normality of an Almost Contact 3-Structure on QR-Submanifolds

We study n-dimensional QR-submanifolds of QR-dimension (p − 1) immersed in a quaternionic space form QP (n+p)/4(c), c ≥ 0, and, in particular, determine such submanifolds with the induced normal almost contact 3-structure.

Naturally reductive homogeneous real hypersurfaces in quaternionic space forms

AbstractWe determine the naturally reductive homogeneous real hypersurfaces in the family of curvature-adapted real hypersurfaces in quaternionic projective space HPn(n ≥ 3). We conclude that the naturally reductive curvature-adapted real hypersurfaces in HPn are Q-quasiumbilical and vice-versa. Further, we study the same problem in quaternionic hyperbolic space HHn(n ≥ 3).