Quaternion Kaehlerian Manifolds Isometrically Immersed in Euclidean Space (original) (raw)

Quaternionic Kaehler manifolds with Hermitian and Norden metrics

2009

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kaehler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kaehlerian and flat are found. It is proved that the quaternionic Kaehler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kaehler quaternionic Kaehler manifold of the considered type is determined.

On Ricci curvature of a quaternion CR-submanifold in a quaternion space form

B. Y. Chen [Kodai Math. J. 4, 399–417 (1981; Zbl 0481.53046)] established a sharp relationship between the Ricci curvature and the squared mean curvature for a submanifold in a Riemannian space form with arbitrary codimension. Recently Ximin Liu [Arch. Math. (Brno), 38, 297–305 (2002; Zbl 1090.53052)] obtained results on Ricci curvature of a totally real submanifold in a quaternion projective space extending the results of Chen. In this article, we wish to estimate the Ricci curvature of a quaternion CR-submanifold in a quaternion space form.

Almost Quaternion-Hermitian Manifolds

Annals of Global Analysis and Geometry, 2000

Following the point of view of Gray and Hervella, we derive detailed conditions which characterize each one of the classes of almost quaternion-Hermitian 4n-manifolds, n > 1. Previously, by completing a basic result of A. Swann, we give explicit descriptions of the tensors contained in the space of covariant derivatives of the fundamental form Ω and split the coderivative of Ω into its Sp(n)Sp(1)-components. For 4n > 8, A. Swann also proved that all the information about the intrinsic torsion ∇Ω is contained in the exterior derivative dΩ. Thus, we give alternative conditions, expressed in terms of dΩ, to characterize the different classes of almost quaternion-Hermitian manifolds. : Primary 53C25; Secondary 53C15, 53C10.

Curvature of almost quaternion-Hermitian manifolds

Forum Mathematicum, 2000

We study the decomposition of the Riemannian curvature R tensor of an almost quaternion-Hermitian manifold under the action of its structure group Sp(n)Sp(1). Using the minimal connection, we show that most components are determined by the intrinsic torsion ξ and its covariant derivative ∇ξ and determine relations between the decompositions of ξ ⊗ ξ, ∇ξ and R. We pay particular attention to the behaviour of the Ricci curvature and the q-Ricci curvature. : Primary 53C55; Secondary 53C10, 53C15.

Generic sub manifold of quaternion Kaehler manifold

International journal of statistics and applied mathematics, 2020

The aim of the present paper is the analysis of a generic sub-manifold in the quaternion Kaehler manifold. Section 1 is the historical background of Kaehler and quaternion manifold. Next, we add some simple formulas and concepts that we use in research work. Again section 3 is about the integrability of generic sub-manifold in quaternion manifold. Further, in the end, area, some fundamental results are obtained concerning the parallel canonical structure in quaternion Kaehler manifold.

A direct approach to quaternionic manifolds

Mathematische Nachrichten, 2016

The recent definition of slice regular function of several quaternionic variables suggests a new notion of quaternionic manifold. We give the definition of quaternionic regular manifold, as a space locally modeled on H n , in a slice regular sense. We exhibit some significant classes of examples, including manifolds which carry a quaternionic affine structure.

Differential geometry of quaternionic manifolds

Annales scientifiques de l'École normale supérieure

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