Hiromasa Tanabe

Bio: Hiromasa Tanabe is an academic researcher from Shimane University. The author has contributed to research in topics: Complex space & Hyperbolic space. The author has an hindex of 3, co-authored 14 publications receiving 31 citations.

Totally geodesic immersions of Kähler manifolds and Kähler Frenet curves

Abstract: In a given Kahler manifold (M,J) we introduce the notion of Kahler Frenet curves, which is closely related to the complex structure J of M. Using the notion of such curves, we characterize totally geodesic Kahler immersions of M into an ambient Kahler manifold and totally geodesic immersions of M into an ambient real space form of constant sectional curvature .

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

Abstract: A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space $$\mathbb {CH}^n$$ . As an application, we study minimal ruled real hypersurfaces in $$\mathbb {CH}^n$$ and also those having constant scalar curvature.

A characterization of the homogeneous ruled real hypersurface in a complex hyperbolic space in terms of the first curvature of some integral curves

Abstract: Due to the works of Berndt and others (J Reine Angew Math 395:132–141, 1989; Geom Dedicata 138:129–150, 2009; Trans Am Math Soc 359:3425–3438, 2007), we find that the class of all homogeneous real hypersurfaces M in $${\mathbb{C}H^n(c)}$$ has just one example which is minimal in this space. Furthermore, in this ambient space, a homogeneous real hypersurface M is minimal if and only if it is ruled. This fact implies that it is interesting to study this minimal homogeneous real hypersurface from the viewpoint of the geometry of ruled real hypersurfaces. It is known that the shape operator A of every ruled real hypersurface M is given by the characteristic field $${\xi}$$ and the vector field U. The purpose of this paper is to characterize this minimal homogeneous real hypersurface in $${\mathbb{C}H^n(c)}$$ in the class of all ruled real hypersurfaces M by investigating the first curvature of all integral curves of the vector fields $${\xi}$$ and U. Note that there exist minimal non-homogeneous ruled real hypersurfaces in $${\mathbb{C}H^n(c)}$$ (see Geom Dedicata 79:267–286, 1999; Hokkaido Math J 43:1–14, 2014).

The homogeneous ruled real hypersurface in a complex hyperbolic space

Abstract: We characterize the homogeneous ruled real hyperurface of a complex hyperbolic space in the class of ruled real hypersurfaces having constant mean curvature.

Generating Curves of Minimal Ruled Real Hypersurfaces in a Nonflat Complex Space Form

Abstract: We first provide a necessary and sufficient condition for a ruled real hypersurface in a nonflat complex space form to have constant mean curvature in terms of integral curves of the characteristic vector field on it. This yields a characterization of minimal ruled real hypersurfaces by circles. We next characterize the homogeneous minimal ruled real hypersurface in a complex hyperbolic space by using the notion of strong congruency of curves.

Hypersurfaces of a Sasakian Manifold

Abstract: We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.

New construction of ruled real hypersurfaces in a complex hyperbolic space and its applications

Abstract: A ruled real hypersurface in a complex space form is a real hypersurface having a codimension one foliation by totally geodesic complex hyperplanes of the ambient space. Our main purpose of this paper is to introduce a new viewpoint to investigate such hypersurfaces in complex hyperbolic space $$\mathbb {CH}^n$$ . As an application, we study minimal ruled real hypersurfaces in $$\mathbb {CH}^n$$ and also those having constant scalar curvature.

Some problems on ruled hypersurfaces in nonflat complex space forms.

Abstract: We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.

Some Problems on Ruled Hypersurfaces in Nonflat Complex Space Forms

Abstract: We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel.