Topics in Almost Hermitian Geometry and Related Fields

PREFACE.

CONTENTS.

Let (M, g) be a Riemannian manifold, TM its tangent bundle and T₁M its unit tangent sphere bundle. The family of all Riemannian g-natural metrics G on TM have been determined by the second author and M. Sekizawa. It depends on 6 arbitrary functions of the norm of a vector u ∈ TM. In the present paper, we study the induced metrics Ḡ on T₁M corresponding to the subfamily of all Riemannian g-natural metrics of the form G = a.g^s + b.g^h + c.g^v, where a, b, c are constants. We calculate the Levi-Civita connection and the standard sorts of curvatures for the above family of metrics. Then, in the three-dimensional case, we derive necessary and sufficient conditions for such metrics to have constant scalar curvature (in terms of the curvature of the underlying manifold). Hereby we generalize some earlier results by E. Boeckx and L. Vanhecke, where the curvature homogeneity of the base manifold plays a crucial role.

We define magnetic mean operators which are generators of random walks under action of magnetic fields. We calculate their formal Taylor expansions and show that the influence of magnetic fields first appears on the third term in general and on the fourth term for Kähler magnetic fields.

In this short note, we give not only a survey of biharmonic submanifolds and biharmonic maps, but also give a Yang-Mills field analogue of biharmonic maps and the Euler-Lagrange equation for the bi-Yang-Mills functional.

We study unit vector fields on a Riemannian manifold (M, g) which embed M as an invariant or anti-invariant submanifold into its unit tangent bundle with respect to its standard contact metric structure.

Recently, V. Oproiu [6,7] introduced a new structure on the tangent bundle, which is defined by a function depending on the energy density on TM. In this article, we study particularly on the induced structure of the unit tangent sphere bundle T₁M as a hypersurface of TM with the new structure. We first show that T₁M admits a contact metric structure. Moreover, we prove that it is K-contact if and only if M is of constant curvature, further in which case the structure is Sasakian.

Bochner’s techniques are used to obtain curvature conditions which ensure that a four-dimensional symplectic or complex pair is a Kähler one in terms of nonnegativity or nonpositivity of sectional curvature.

For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish.

No abstract received.

The aim of this paper is to give examples of compact neutral 4-manifolds (M, g) whose Ricci tensor ρ satisfies the relation . We present also a family of new Einstein bi-Hermitian neutral metrics on ruled surfaces of genus g > 1.

For every affine crystallographic subgroup Γ of Aff(n), L. S. Auslander conjectured that Γ is virtually solvable. The purpose in the present paper is to find an algebraic condition on Γ which leads the affirmative answer to the conjecture. The author considers the proper discontinuity of virtually torsionfree subgroups of Aff(n) by dividing into several types as for the Zariski closure of the linear part. This method leads a partial answer of Auslander’s conjecture for n ≦ 6.

Let M be a real hypersurface with almost contact metric structure (ϕ, ξ, η, g) in a non-flat complex space form M_n(c). In this paper we investigate real hypersurfaces of M_n(c) whose structure Jacobi operator R_ξ commutes with both the structure tensor ϕ and the Ricci tensor S of M. We characterize Hopf hypersurfaces of M_n(c).

In this paper we give a survey on characterizations of totally η-umbilic real hypersurfaces, ruled real hypersurfaces and Hopf hypersurfaces with constant principal curvatures in a complex projective or complex hyperbolic space by the extrinsic shape of geodesics on these real hypersurfaces.

The tangent bundle as a 4n-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of manifold are received from the known explicit examples in that manner.

It is well-known that an almost complex structure J is defined on the 6-dimensional unit sphere S⁶ by using the multiplication of the Cayley algebra 𝔈. The group G₂ of automorphisms of the Cayley algebra acts on S⁶ transitively and we have a diffeomorphisms S⁶ = G₂/SU(3). The action of G₂ on S⁶ extends to the action on the Grassmann bundle consisting of oriented 3-dimensional tangent subspaces.

In this note we consider 3-dimensional submanifolds of S* such that the tangent space at any point is contained in an orbit of the action of G₂ on .

The set of orbits of the action of G₂ on is parametrized by a complex number λ (|λ| ≤ 1) (see Proposition 2). We denote by Σ_λ the orbit which corresponds to λ and call a 3-dimensional submanifold N a Σ_λ-submanifold if and only if T_xN ∈ Σ_λ for any x ∈ N.

The aim of this paper is to show the non-existence of compact Σ_λ-submanifold for λ ∉ R.

The purpose of the present paper is to formulate a Fourier transformation for the space of functionals, as an infinitesimal meaning. We extend the real numbers R to *(*R) on the basis of nonstandard methods for the construction. The domain of a functional is the set of all internal functions from a *-finite lattice to a *-finite lattice with a double meaning. Considering a *-finite lattice with a double meaning, we propose a new treatment of the domains for functionals in the theory of Fourier transformation, and exhibit two typical examples.

No abstract received.

We investigate the α-family of almost Kähler structures on the tangent bundle TS over a statistical model S. We show that TS becomes a Kähler manifold of constant holomorphic sectional curvature in particular cases.

In this survey we consider the notion of ξ-invariant or 𝔇^⊥-invariant real hypersurfaces in complex two-plane Grassmannians G₂(ℂ^m+2) and give a complete classification for real hypersurfaces satisfying these both two conditions. On the other hand, we prove that there do not exist any real hypersurfaces in G₂(ℂ^m+2) with parallel second fundamental tensor and give some generalizations of such situations for real hypersurfaces in G₂(ℂ^m+2) with parallel second fundamental tensor on a distribution 𝔉 defined by 𝔉 = ξ ∪ 𝔇^⊥, where 𝔇^⊥ = Span {ξ₁, ξ₂, ξ₃}. Related to this topics we introduce some works due to Berndt and Suh [4], Suh [14] for the characterizations of type A with isometric Reeb flow or vanishing Lie derivative. Finally we introduce a new characterization of type B in the paper due to Suh [15].

The moduli space of null-correlation bundles over ℂP²ⁿ⁺¹ has a natural L₂-Kähler metric. We describe explicitly the L₂-Kähler metrics of the moduli spaces and respectively. We also obtain that the L₂-Kähler metrics of these moduli spaces can be written down in terms of the Gaussian hypergeometric functions and the Appell’s hypergeometric functions of type F₁ respectively.

No abstract received.

No abstract received.