Kouhei Miura | Miyagi University (original) (raw)

Papers by Kouhei Miura

International Electronic Journal of Geometry

Tohoku Mathematical Journal

We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in ... more We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in pseudo-Riemannian space forms of neutral signature. As a corollary, a characterization theorem is obtained for the Lorentzian Boruvka spheres, that is, a full real analytic null r-planar geodesic immersion with vanishing mean curvature vector field is locally congruent to the Lorentzian Boruvka sphere in a 2r-dimensional space form of neutral signature. 1. Introduction. To study minimal surfaces in a unit sphere, the twistor lift plays an important role. For instance, Calabi [2] proves a rigidity theorem for minimal immersions of surfaces with genus zero in Euclidean spheres using twistor lifts. An application of the rigidity result shows that a minimal isometric immersion of the 2-sphere into a unit sphere is congruent to the d-th standard immersion (also called the Boruvka sphere in the unit 2dsphere) for a positive integer d. As a result, Boruvka spheres have horizontal twistor lifts. Chern [4] reinterprets Calabi's work and investigates minimal 2-spheres in a unit sphere by using the higher order osculating spaces and higher fundamental forms. We refer to Bryant [1] also. One of the aims in this paper is to characterize Boruvka spheres in indefinite pseudo-Riemannian geometry, using an indefinite version of twistor lifts. The Boruvka spheres with Lorentzian metric, a family of isometric immersions of Lorentzian 2-sphere into the pseudo-Riemannian spheres are, via Wick rotations, constructed from the standard immersions of Riemannian 2-sphere in Ding and Wang [5] and Miura [10]. These immersions have vanishing mean curvature vector fields, thus, these are extremal. In this paper, we call these immersions the Lorentzian Boruvka spheres (LBSs). We focus on the fact that the target spaces of the LBSs are always neutral. Then it is natural that we use reflector lifts instead of twistor lifts. The notion of reflector lifts established on neutral pseudo-Riemannian manifolds is corresponding to that of twistor lifts in Riemannian geometry. See Jensen and Rigoli [7] for details. We see that extremal helical geodesic immersions (HGIs) from Lorentzian surfaces into a space form have horizontal reflector lifts. Note that the LBSs have helical geodesics. As a property of HGIs, we propose a notion of null r-planar geodesic immersions (PGIs). For a precise definition, see Definition 4.1. We provide a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts. As an application of our

The Michigan Mathematical Journal

Tokyo Journal of Mathematics, 2008

We study an isometric immersion f : M →M with geodesic normal sections, whereM is a semi-Riemanni... more We study an isometric immersion f : M →M with geodesic normal sections, whereM is a semi-Riemannian space form. In Riemannian geometry, it is known that f is helical, in particular, every geodesics of M have the same proper order inM. However this does not hold in general, whenM is indefinite semi-Riemannian. We give sufficient conditions for an isometric immersion with geodesic normal sections to be helical.

Results in Mathematics, 2009

Kodai Mathematical Journal, 2007

We obtain some basic results on helical geodesic immersions in semi-Riemannian geometry. For exam... more We obtain some basic results on helical geodesic immersions in semi-Riemannian geometry. For example, it is shown that, for an indefinite semi-Riemannian submanifold, if any space-like geodesics of the submanifold are helices of order d, curvatures l 1 ;. .. ; l dÀ1 and signatures e 1 ;. .. ; e d in the ambient space, then any timelike geodesics of the submanifold have the same order and curvatures, and signatures ðÀ1Þ 1 e 1 ;. .. ; ðÀ1Þ d e d .

Tsukuba Journal of Mathematics