Recent Topics in Differential Geometry and its Related Fields

The following sections are included:

• PREFACE
• ORGANIZING, LOCAL AND SCIENTIFIC ADVISORY COMMITTEES
• PRESENTATIONS
• CONTENTS

Professor Stancho Dimiev was a member of the organizers of the conference “The International Workshop on Complex Structures and Vector Fields”, which is the former academic program on Analysis and Geometry between Bulgaria and Japan. We received news that he had passed away on January 30, 2018. We here devote several pages in memoriam of him, who was one of the founders of our academic exchanging program…

We obtain new invariant Einstein metrics on the compact Lie group SU(2n) (n ≥ 3) which are not naturally reductive. We view the compact Lie group SU(2n) as total space over the generalized Wallach space SU(2n)/U(n) and consider Ad(U(n))-invariant metrics on the group SU(2n) associated to the generalized Wallach space to obtain these invariant Einstein metrics.

In this paper, we construct noncompact homogeneous Lagrangian submanifolds in some Hermitian symmetric spaces of noncompact type. Our noncompact homogeneous Lagrangian submanifolds are obtained as connected closed subgroups of the solvable part of the Iwasawa decomposition.

We obtain a characterization of complex curves in Kähler surfaces. In addition, by a direct method, we see that in a hyperKähler 4-manifold, complex curves are just isotropic minimal surfaces compatible with the orientation of the space. We obtain characterizations of hyperKähler 4-manifolds among Kähler surfaces, in terms of isotropic minimal surfaces and deformations of the complex structures.

Our purpose here is to give a characterization à la Etayo and Santamaria for the general natural (α, ε)-structures on the total space T*M of the cotangent bundle of a Riemannian manifold (M, g). We first obtain the general natural α-structures on T*M, and prove that they depend on four coefficients of the energy density t in a cotangent vector p. Moreover, we show that the α-structures obtained on T*M are integrable if and only if (M, g) is of constant sectional curvature c, and three of the coefficients depend on the other coefficients, their derivatives, c, t and α. Then, we consider a general natural lifted metric G on T*M, and by studying its ε-compatibility with the obtained α-structures, we characterize the two classes of general natural (α, ε)-structures on T*M (according to the value 1 or −1 of the product αε). We show that for the class with αε = −1 there are two proportionality relations between the coefficients of the α-structure and those of the metric. We prove that in this case the fundamental tensor field Ω is a closed 2-form (i.e. the structure is of (almost) Kähler type) if and only if one of the proportionality factors is the derivative of the other one.

A Kähler graph consists of two graphs having the same set of vertices. When these two graphs are regular and their adjacency operators are commutative, the probabilistic adjacency operators of derived graphs are selfadjoint. In this paper we study maximum eigenvalues of derived graphs of such a regular Kähler graph whose constituent graphs are connected, and investigate whether derived graphs are connected or not.

We introduce here the notion of semi-Riemannian D-general warping (M = M₁ × M₂, g) by extending two geometric notions, one defined by Blair and the other by Tanno. On these kinds of manifolds, we study the behavior of geodesics, magnetic curves and F-geodesics. Then on a semi-Riemannian D-general warping, we investigate some special vector fields, such as Killing, concircular, concurrent and recurrent.

We introduce the notion of a duality between compact symmetric triads and semisimple pseudo-Riemannian symmetric pairs, which is a generalization of that of the duality between compact/non-compact Riemannian symmetric pairs. Moreover we state a relation between compact symmetric triads and symmetric triads with multiplicities. As its application we state an outline of an alternative proof of Berger’s classification of pseudo-Riemannian symmetric pairs. Moreover, we also give an alternative proof of Leung’s classification of reflective submanifolds in compact Riemannian symmetric spaces and that of real forms in compact Hermitian symmetric spaces.

In this paper we clarify a property of the complex vector space of holomorphic vector fields on a simple irreducible pseudo-Hermitian symmetric space by means of the representation theory.

The purpose of this paper is to give some fibre bundle structures related to the special unitary group SU(4). The classical Lie group SU(4) is isomorphic to the spinor group Spin(6) which is a double covering group of the special orthogonal group SO(6): This isomorphism gives rise to some fibre bundle structures on some homogeneous spaces related to SU(4): By using this structure, we give a relationship between a certain non-at totally geodesic surface in SU(4)/SO(4) and in Sp(2)/U(2) ≃ Spin(5)/U(2).

We consider the problem of computing the weight distribution of a linear code of dimension k over a composite finite field ${\mathbb{F}}_q$ where q = 2^m. Due to the trace function we reduce the arithmetic operations over the composite field to those over a prime field. This allows us to apply a transform of Walsh-Hadamard type. The codes are represented by their characteristic vector with respect to a given generator matrix and a generator matrix of the k-dimensional simplex code $S_{q:k}$. The developed algorithm has complexity O(kmq^k−1).

We study the property that trajectories on real hypersurfaces of type (B) in a complex hyperbolic space are tangentially of order two. By investigating the image of the moduli space of such trajectories in the moduli space of circles on a complex hyperbolic space, we give some characterizations of homogeneous Hopf hypersurfaces.

In this paper, we give a very simple method to construct closed plane curves. Furthermore, we prove that the space of curvature functions of closed plane curves is dense in the space of periodic continuous functions on ℝ.