Modern Approaches to Differential Geometry and its Related Fields

The following sections are included:

• PREFACE
• PRESENTATIONS
• CONTENTS

In this paper, we show the existence of complete Z_N -invariant conformal minimal immersions from finitely punctured real projective planes into R³ with various types of ends. In particular, there exists such a surface with N embedded ends and 1 non-embedded (or embedded) end for any d = deg g ≥ 2N + 1 if gcd(d, N ) = 1.

Two special neutral hypersurfaces ℒ_± in the light cone $L\left({\bigwedge }^2{E}_1^4\right)$ studied in [1], [2] are SO(3, 1)-orbits. In this paper, we see that each SO(3, 1)-orbit in $L\left({\bigwedge }^2{E}_1^4\right)$ is either a neutral hypersurface homothetic to one of ℒ_± in $L\left({\bigwedge }^2{E}_1^4\right)$ or a hypersurface with a two-dimensional involutive distribution where the induced metric vanishes. The difference between these hypersurfaces can be understood in terms of the stabilizer and the r-slice of $L\left({\bigwedge }^2{E}_1^4\right)$ for r > 0.

In this paper, we study Einstein solvmanifolds constructed from parabolic sub-algebras of real special linear Lie algebras. We calculate sectional curvatures of these Einstein solvmanifolds, and show that they do not have nonpositive sectional curvature unless they are not isometric to symmetric spaces.

We consider invariant Einstein-like metrics of type ℬ, that is, Ricci tensor r is a Codazzi tensor, on generalized flag manifolds. We show that, for generalized flag manifolds G/K with second Betti number b₂(G/K) = 2 and with ν isotropy summands (ν ≤ 5) or G₂-type, then invariant Einstein-like metrics of type ℬ are Einstein.

The product S³ × S⁵ of 3-dimensional and 5-dimensional spheres is a complex manifold, which is defined by Calabi-Eckmann ([2]). This complex structure comes from the fact that S³ ×S⁵ is a total space of a T ²-bundle over ℂℙ¹ ×ℂℙ², with a natural projection map π₁ × π₂ : S³ × S⁵→ ℂℙ¹ × ℂℙ². Let $W_1=ℂ{\mathbb{P}}^2\text{\#}\overline{ℂ{\mathbb{P}}^2}$ be one of Hirzebruch surfaces which is obtained as a blowing up of ℂℙ² at one point. This surface realizes as a complex submanifold in ℂℙ¹ ×ℂℙ². Then the inverse image ${\tilde{W}}_1={\left({\pi }_1\times {\pi }_2\right)}^{-1}\left(W_1\right)$ of W₁ is a complex submanifold in S³ × S⁵. First we shall show that ${\tilde{W}}_1$ is diffeomorphic to S³ × S³ concretely. By using this diffeomorphism, we give a representation of the induced complex structure J of ${\tilde{W}}_1$ from S³ × S⁵, as a (1, 1) tensor field on ${\tilde{W}}_1$ with respect to the left invariant vector fields on S³ × S³. We note that the induced complex structure J is different from the (Calabi-Eckmann type) complex structure of a direct Riemannian product S³ × S³, which is obtained as a total space of a T ² bundle over ℂℙ¹ × ℂℙ¹. Also we describe a blowing up of ℝℙ² at one point, as a Lagrangian surface of W₁. Then this Lagrangian surface is a Klein bottle, which can be considered as a connected sum ℝℙ²#ℝℙ² in $ℂ{\mathbb{P}}^2\text{\#}\overline{ℂ{\mathbb{P}}^2}$.

We study whether trajectories for Kähler magnetic fields on a complex hyperbolic space are obtained as extrinsic shapes of trajectories on some real hypersurface of type (A). We show that bounded trajectories are expressed by geodesics and by trajectories on some real hypersurfaces but unbounded trajectories are not expressed by them if strengths of Kähler magnetic fields are small compared with the absolute value of the holomorphic sectional curvature.

F-Yang-Mills connections are critical points of the F-Yang-Mills functional defined on the space of connections of a principal fiber bundle. Here, F is a nonnegative strictly increasing C²-function. In this paper, we study the instability of the F-Yang-Mills connections. We derive a sufficient condition that any non-flat, F-Yang-Mills connection over the standard sphere is unstable.

This is an announcement of a not-yet-published paper entitled “A Simons type condition for instability of F-Yang-Mills connections” jointwork with Kazuto Shintani (arXiv:2301.0429).

In this paper, we present an overview of some combinatorial objects having interesting structure, namely, association schemes and strongly regular graphs. We propose an approach to obtain a certain type of association schemes from two weight linear codes. We use the projective dual transform of linear codes, and its basic concept is also described in the article.

A Kähler graph consists of two kinds of graphs having the same set of vertices. As candidates of model Kähler graphs which correspond to homogeneous spaces admitting magnetic fields, vertex transitive normal Kähler graphs are proposed in [5]. To give many examples of such Kähler graphs, we define some product operations. In this note, we restrict ourselves to the product Kähler graphs whose first kind graphs are of tensor product type, and study whether they are connected, are bipartite, are vertex-transitive and are normal.

In this paper, we study biconservative hypersurfaces $M_r^4\hspace{0.17em}(r=0,1,2,3,4)$ in the pseudo-Euclidean space ${\mathbb{E}}_s^5\hspace{0.17em}(s=0,1,2,3,4,5)$ with constant norm of second fundamental form. We found that every such hypersurface in ${\mathbb{E}}_s^5$ with diagonal shape operator has constant mean curvature and constant scalar curvature.

In this paper, we deal with the geometry of orbits of hyperpolar actions, which include Hermann actions, on compact Riemannian symmetric spaces.

In [30], Tachibana introduced the notion of ^*-Ricci tensor on almost Hermitian manifolds. Motivated by the previous work, Hamada [18] defined ^*-Ricci tensors of real hypersurfaces in non-flat complex space forms. Using ^*-Ricci tensors, in [20], the authors introduced the notion of ^*-Ricci solitons of real hypersurfaces in non-flat complex space forms. This short note aims to present the progress in the study of ^*-Ricci soliton.

In information geometry, the geometry of statistical models is primarily constructed based on the criteria of invariance under Markov embeddings and dual flatness for a pair of affine connections. For exponential families, invariance and dual flatness hold simultaneously. On the other hand, for non-exponential families, these criteria do not hold simultaneously. In this paper, after reviewing the invariant geometry of statistical models including Markov embeddings, an invariant and dually flat structure is constructed for several q-exponential families.

The following section is included:

• AFTERWORD