Prospects of Differential Geometry and Its Related Fields

The following sections are included:

• PREFACE

• ORGANIZING AND SCIENTIFIC ADVISORY COMMITTEES

• PRESENTATIONS

• CONTENTS

This article explains our recent contributions to studies of biharmonic maps concentrated to three topics, (1) our recent results on L²- rigidity of biharmonic maps with S. Gudmundsson, N. Nakauchi, on the generalized Chen’s conjecture (cf. [16]), (2) some classification of biharmonic Lagrangian submanifolds in complex space form generalizing Sasahara’s results (cf. [21]) due to a joint work with Sh. Maeta (cf. [18]), and (3) conformal change of metrics and biharmonic maps due to a joint work with H. Naito (cf. [19]).

We construct the Einstein equation for an invariant Riemannian metric on generalized flag manifolds G/K with G₂-type t-roots. By computing a Gröbner basis for a system of polynomials on six variables, we prove that such a generalized flag manifold G/K, which is not the full flag manifold G₂/T, admits exactly one invariant Kähler Einstein metric and six non Kähler invariant Einstein metrics up to isometry and scalar.

Codes over Gaussian integers are suitable for coding over two-dimensional signal space. We present a metric over QAM-like constellations modeled by quotient rings of Gaussian integers. Moreover, we give some simple examples for codes correcting errors of Mannheim weights one and two.

This is a survey on generalized Liouville manifolds which were studied in the joint work with K. Sugahara ([3]).Without assuming that the metric is positive definite, we define a generalized Liouville structure and generalized Liouville manifolds whose geodesic flows are completely integrable. We give many generalized Liouville manifolds and give a classification of them.

Einstein’s program of the unified field theory transformed nowadays to the “theory of everything” (TOE) requiring new primordial elements and relations between them. Definitely, they must be elements of the quantum nature. One of most fundamental quantum elements are pure quantum states. Their basic relations are defined by the geometry of the complex projective Hilbert space. In the framework of such geometry all physical concepts should be formulated and derived in the natural way. Analysis following this logic shows that inertia and inertial forces are originated not in spacetime but in the space of quantum states since they are generated by the deformation of quantum states as a reaction on an external interaction or self-interaction. It turns out that the ℂP^N–1 geometry intrinsically connected with such deformations. Therefore, such geometry should clarify the old problem of inertial mass (dynamical mass generation), and, on the other hand, leads to the law of environment independence (invariance) of the quantum numbers of quantum particles. The conservation law of proper energy-momentum following from this invariance has been applied to self-interacting quantum Dirac’s electron.

This article is a survey of recent results on LeBrun-Mason type twistor theory. We summarize two types of results given by the author, one is the twistor theory for Tod-Kamada metric and the other is its analogy on ℝ⁴. We introduce several integral transforms to write down these correspondences explicitly. We obtain consequences concerning solutions to wave equations by making use of these integral transforms. A general method of twistor construction on the indefinite geometry is also summarized.

In this note we attempt to reconsider circles of positive geodesic curvature on a complex hyperbolic space by use of magnetic fields on real hypersurfaces. In particular, we show that every horocyclic circle is a trajectory for some Sasakian magnetic field on a horosphere. We show also that the same property holds for horocyclic helices of proper order 3 each of which is an orbit of one parameter family of isometries and lies on a totally geodesic complex hyperbolic plane.

In this paper we study balls and spheres formed by trajectories for Kähler magnetic fields on a Kähler manifold and give estimates of their volume elements under some assumptions on sectional curvatures.

On the projectively extended euclidean plane of a fixed triangle, point-wise binary operations and transformations described in barycentric coordinates arise in a geometric way. We apply these operations over the points of a standard Bézier parabola and a weigh point, and get families of associated curves. There exist two geometric multiplicative group structures. We describe the trajectories of a rational Bézier curve by means of group orbits of its points. We calculate the curvature of a smooth curve in barycentrics and apply the formula to a rational Bézier curve.

We give the geometrical structures related to the orthogonal homogeneous almost complex structure on S³ × S³ which is obtained by the multiplication of the octonions ℭ.

In this paper, we report on a moduli theory of compact oriented minimal surfaces in flat tori. A key space of the moduli space can be classified into connected components via the Morse index and nullity, and some connected components have good geometric structures. Moreover, the theory includes a procedure to compute the Morse index of a minimal surface with only trivial Jacobi fields. As its applications, we can compute the Morse index of some examples of minimal surfaces. This is a survey of the previous works [3–5].

For curves in the purely imaginary octonions, we represent their curvatures by using some of their G₂-invariant functions.

We give a survey on geometry of statistical manifolds in terms of estimating functions. A statistical model naturally has a statistical manifold structure. In particular, a q-exponential family which is a generalization of an exponential family admits several statistical manifold structures. An estimating function can be regarded as a tangent vector of a statistical model, and it gives rise to dualistic structures on a statistical manifold. In this paper, we construct statistical manifold structures on statistical models and divergence functions from the viewpoint of estimating functions. We also study geometry of nonintegrable estimating functions.

A Hessian domain is a flat statistical manifold, and its level surfaces are 1-conformally flat statistical submanifolds. In addition, a level surface with an α-connection is α-conformally equivalent to a statistical manifold with an α-transitively flat connection for each real number α. In this study, we investigate these facts and present details related to α-connections.

We construct examples of ruled complex Lagrangian submanifolds in C⁴ as deformations of the complex Lagrangian cone in C⁴.

The following sections are included:

• POSTFACE