Contemporary Perspectives in Differential Geometry and its Related Fields

The following sections are included:

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

We obtain new invariant Einstein metrics on the Stiefel manifolds V_mℂ^m+n = SU(m+n)/SU(n) and the compact Lie group SU(3+n) (n ≥ 5) which are not naturally reductive. These are Ad(S(SO(m) × U(1) ×U(n)))-invariant metrics and Ad(S(SO(3) × U(1) × U(n)))-invariant metrics respectively.

Let M be a Riemannian symmetric space of compact type, and we denote by G the identity component of the isometry group Isom(M) of M. In this paper, we give an algorithm to classify G-conjugacy classes of geodesics in a given reflective submanifold of M in terms of Satake diagrams.

We construct the canonical ℝ-mixed Hodge structure on Sullivan’s 1-minimal mode of the de Rham complex of a compact Kähler manifold. Unlike the Hodge structure on the cohomology of a compact Kähler manifold, such ℝ-mixed Hodge structure depends on the choice of a Kähler metric and functorial for holomorphic isometric maps.

We introduce here the notion of distorted Riemann extension as a generalization of classical Riemann extension. We construct a class of semi-Riemannian metrics of neutral signature on the total space of the cotangent bundle of a manifold endowed with Weyl structure and we obtain an equivariance result. Then we provide a class of F-geodesics (which generalize both classical geodesics and magnetic curves) on T*M, with respect to this distorted Riemann extensions.

The notion of Hermann type actions is an analogy of that of Hermann actions. Hermann type actions are defined in the pseudo-Riemmanian category while Hermann actions are defined in the Riemannan category. It is known that Hermann actions have nice properties such as hyperpolarity. We study some nice properties of Hermann type actions like hyperpolar actions in the Riemannian category.

The homogeneous space SU(2)/SO(2) can be considered as the Grassmaniann manifold of all totally real 2-planes of the complex plane ℂ². Also the homogeneous space SU(2)/S(U(1) × U(1)) can be considered as the Grassmaniann manifold of all complex lines of ℂ². There does not exist a canonical isomorphism from a totally real 2-plane to a complex line, though, they are isomorphic to a 2-dimensional real vector space. Nevertheless, we can give the isomorphism from SU(2)/SO(2) to SU(2)/S(U(1)×U(1)). The purpose of this paper is to give this isomorphism and the bundle map from SU(2) to SU(2) corresponding to this isomorphism related to the Hopf fibrations, explicitly. By using this isomorphism we give some applications of non-flat totally geodesic immersions from S² to symmetric spaces of Type A.

A trajectory-horn associated with a geodesic is a variation of trajectories for a Kähler magnetic field where each of the trajectories joins the origin and another point of the geodesic. We study lengths of these trajectory-segments, inner products of the tangent vectors of the geodesic and trajectories, and angles of two trajectories at the origin of the geodesic. By comparing trajectory-horns with those on complex space forms, we estimate these quantities under an assumption that sectional curvatures are bounded from above.

In this paper we study null curves on the unit tangent bundle U(TM) with a timelike unit normal vector field of a 2-dimensional Kähler-Norden manifold M. We find relationships between curves on the base manifold M and their corresponding null curves on U(TM), which is a 3-dimensional Lorentzian manifold. The objects of investigation are also Legendre curves on U(TM) with respect to the almost contact B-metric structure. We show that the considered Legendre null curves are Cartan framed null curves with respect to the original parameter. We also give some examples of such curves.

We study Hopf tori obtained from a Viviani curve, small circles and a great circle in S². By using expressions of horizontal lifts of these curves we give parametrizations of these Hopf tori, and show some geometric properties.

The 3rd homotopy groups of homogeneous spaces G₂/S_p(1)_± and G₂/U(2)_± are determined. Consequently we see that G₂/S_p(1)₊ and G₂/S_p(1)₋ are not homeomorphic to each other, and similarly for G₂/U(2)₊ and G₂/U(2)₋. The key is the notion of Dynkin index, which is also explained.

We give a survey on sequential structures of escort distributions and sequential structures of statistical manifolds. For a deformed exponential family, the notion of escort distribution has been introduced in nonextensive statistical physics, and an expectation with respect to its escort distribution has been discussed. The author introduced a sequential structure of escort distributions, and discussed statistical manifold structures derived from sequential escort distributions. In this paper, after reviewing basic facts on information geometry, we study sequential structures of statistical manifolds. A divergence function which is a distance-like function on a statistical manifold plays an important role in this framework.