Toshiaki Adachi

Geometry and Topology

23.25

Publications

  • Toshiaki Adachi, Tuya Bao, Sadahiro Maeda
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    ABSTRACT: In a non-flat complex space form ℂM n (c) of constant holomorphic sectional curvature (c≠0) and of complex dimension n (≥2) there are two types of real hypersurfaces. One is the class of Hopf hypersurfaces while the second is the class of ruled real hypersurfaces. The present paper is devoted to a classification of minimal ruled real hypersurfaces in correspondence with a classification of totally real circles on non-flat complex space forms. As example, it is obtained that the maximal minimal ruled real hypersurfaces of ℂP n are not complete and they are congruent to each other with respect to the action of its isometric group.
    Hokkaido Mathematical Journal 02/2014; 43(1). DOI:10.14492/hokmj/1392906097 · 0.26 Impact Factor
  • SADAHIRO MAEDA, TOSHIAKI ADACHI, YOUNG HO KIM
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    ABSTRACT: Totally η-umbilic real hypersurfaces are the simplest examples of real hypersurfaces in a non-flat complex space form. Geodesic hyperspheres in this ambient space are typical examples of such real hypersurfaces. We characterise every geodesic hypersphere by observing the extrinsic shapes of its geodesics and using the derivative of its contact form.
    Glasgow Mathematical Journal 01/2013; 55(01). DOI:10.1017/S0017089512000456 · 0.31 Impact Factor
  • Tuya Bao, Toshiaki Adachi
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    ABSTRACT: We study circular trajectories for Sasakian magnetic fields on geodesic spheres, horospheres and tubes around totally geodesic complex hypersurfaces in a complex hyperbolic space. Investigating their extrinsic shapes in the ambient complex hyperbolic space, we give conditions for them to be bounded and to be closed. By use of information on lengths of circles in complex space forms, we give expressions of lengths of circular trajectories on those real hypersurfaces and show that their length spectrum is a discrete subset of a real line.
    Kodai Mathematical Journal 10/2011; 34(2011). DOI:10.2996/kmj/1320935555 · 0.35 Impact Factor
  • Tuya Bao, Toshiaki Adachi
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    ABSTRACT: On a real hypersurface in a Kähler manifold we can consider a natural closed 2-form associated with the almost contact metric structure induced by Kähler structure. We treat trajectories under magnetic fields which are constant multiples of this 2-form. We consider a condition for them to be also curves of order 2 on tubes around totally geodesic real hyperbolic spaces in a complex hyperbolic space.
    Differential Geometry and its Applications 08/2011; 29. DOI:10.1016/j.difgeo.2011.04.004 · 0.59 Impact Factor
  • Toshiaki Adachi
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    ABSTRACT: On a Kähler manifold we have natural uniform magnetic fields which are constant multiples of the Kähler form. Trajectories, which are motions of electric charged particles, under these magnetic fields can be considered as generalizations of geodesics. We give an overview on a study of Kähler magnetic fields and show some similarities between trajectories and geodesics on Kähler manifolds of negative curvature.
    Differential Geometry and its Applications 08/2011; 29. DOI:10.1016/j.difgeo.2011.04.001 · 0.59 Impact Factor
  • Toshiaki ADACHI, Masumi KAMEDA, Sadahiro MAEDA
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    ABSTRACT: In an $n$ $(\geqq2)$-dimensional nonflat complex space form $\widetilde{M}_n(c)(=\mathbb{C}P^n(c)$ or $\mathbb{C}H^n(c)$), we classify real hypersurfaces $M^{2n-1}$ which are contact with respect to the almost contact metric structure $(\phi,\xi,\eta,g)$ induced from the K\"ahler structure $J$ and the standard metric $g$ of the ambient space $\widetilde{M}_n(c)$. Our theorems show that this contact manifold $M^{2n-1}$ is congruent to a homogeneous real hypersurface of $\widetilde{M}_n(c)$.
    Hokkaido Mathematical Journal 06/2011; 40(2011). DOI:10.14492/hokmj/1310042828 · 0.26 Impact Factor
  • Tuya Bao, Toshiaki Adachi
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    ABSTRACT: We study trajectories for Sasakian magnetic fields which are also circles of positive geodesic curvature on geodesic spheres in a complex projective space. Investigating their extrinsic shapes we give a condition for them to be closed. By use of information on lengths of circles on a complex projective space, we give their lengths, and estimate the bottom of the length spectrum of circular trajectories.
    Kodai Mathematical Journal 06/2011; 34(2011). DOI:10.2996/kmj/1309829549 · 0.35 Impact Factor
  • Toshiaki Adachi, Masumi Kameda, Sadahiro Maeda
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    ABSTRACT: We show that M2n-1 is a real hypersurface all of whose geodesics orthogonal to the characteristic vector ξ are mapped to circles of the same curvature 1 in an n-dimensional nonflat complex space form $\widetilde{M}_n$ (c) (= CPn(c) or CHn(c)) if and only if M is a Sasakian manifold with respect to the almost contact metric structure from the ambient space $\widetilde{M}_n$ (c). Moreover, this Sasakian manifold M is a Sasakian space form of constant φ-sectional curvature c + 1 for each c (≠0).
    Kodai Mathematical Journal 10/2010; 33(2010). DOI:10.2996/kmj/1288962549 · 0.35 Impact Factor
  • Sadahiro Maeda, Toshiaki Adachi
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    ABSTRACT: We characterize some real hypersurfaces in an n-dimensional nonflat complex space form Mn(c)( = \mathbbCPn(c)or\mathbbCHn(c))M_{n}(c)(= {\mathbb{C}}P^{n}(c)\,\, {\rm or}\, \, {\mathbb{C}}H^{n}(c)) in terms of Sasakian curves on real hypersurfaces which are closely related to their almost contact metric structures induced from the ambient space Mn(c). We also classify curves on a geodesic sphere of \mathbbCPn(c){\mathbb{C}}P^{n}(c) which are mapped to circles on some standard sphere through the well-known isometric embedding, and show that these curves are Sasakian curves on this geodesic sphere. Mathematics Subject Classification (2000).Primary 53C40-Secondary 53C22 Keywords.Nonflat complex space forms-real hypersurfaces-hypersurfaces of type (A)-Sasakian curves-structure torsion-normal curvature-geodesic spheres-the first standard minimal embedding-Euclidean spheres-homogeneous curves
    Results in Mathematics 12/2009; 56(1):489-499. DOI:10.1007/s00025-009-0400-2 · 0.64 Impact Factor
  • Tuya Bao, Toshiaki Adachi
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    ABSTRACT: In this paper we study which trajectories for Sasakian magnetic fields are circles on certain standard real hypersurfaces which are called hypersurfaces of type A in a nonflat complex space form. We also give a characterization of these real hypersurfaces by such a circular property of trajectories for Sasakian magnetic fields. Mathematics Subject Classification (2010)Primary 53C40-Secondary 53C22 KeywordsNonflat complex space forms-real hypersurfaces-Sasakian magnetic fields-circles
    Journal of Geometry 01/2009; 96(1):41-55. DOI:10.1007/s00022-010-0032-4
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    Toshiaki Adachi
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    ABSTRACT: In this note we study a foliation on the moduli space of helices of order not greater than 4 on a real space form which corresponds to length spectrum.
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    Sadahiro MAEDA, Toshiaki ADACHI, Young Ho KIM
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    ABSTRACT: It is well-known that there exist no homogeneous ruled real hypersurfaces in a complex projective space. On the contrary there exists the unique homogeneous ruled real hypersurface in a complex hyperbolic space. Moreover, it is minimal. We characterize geometrically this minimal homogeneous real hypersurface by properties of extrinsic shapes of some curves.
    Journal of the Mathematical Society of Japan 01/2009; 61(2009). DOI:10.2969/jmsj/06110315 · 0.64 Impact Factor
  • Yusuke Takeo, Toshiaki Adachi
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    ABSTRACT: We take a convex pentagon R=ABCDE satisfying AB ¯=BC ¯, AE ¯=ED ¯ and ∠B=∠E=π/2. By use of 4 pentagons congruent to R we make a fundamental hexagon and give a tessellation of a Euclidean plane, which is called a tessellation of tiling-type 4 by congruent pentagons. This tessellation has 3-valent and 4-valent vertices. We study Dirichlet property for such tessellations of a plane. We also consider similar tessellations by congruent original and reversed pentagons which have fundamental regions made by 4 pieces and study their Dirichlet property.
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    Yusuke Takeo, Toshiaki Adachi
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    ABSTRACT: In this paper we give a necessary and sufficient condition for some tessellations of a plane R 2 by congruent convex quadrangles to be Dirichlet.
    01/2009; 39.
  • Toshiaki Adachi
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    ABSTRACT: In this paper we study some basic properties of trajectories for canonical magnetic fields induced by structure tensor on real hypersurfaces of types A0 and A1 in a complex space form. On each such real hypersurface, there are infinitely many canonical magnetic fields whose trajectories with null structure torsion are closed, and also infinitely many canonical magnetic fields whose trajectories with null structure torsion are open. We give a condition dividing canonical magnetic fields into these two classes and investigate lengths of closed trajectories. Our study also provide us information on the moduli space of Killing helices of proper order 4 on complex space forms.
    Journal of Geometry 12/2008; 90(1):1-29. DOI:10.1007/s00022-008-1941-3
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    Toshiaki Adachi, Tadashi Sugiyama
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    ABSTRACT: In this paper we show that an isometric immersion is isotropic in the sense of O'Neill if and only if it preserves logarithmic derivatives of first geodesic curvatures of some curves.
    Differential Geometry and its Applications 06/2008; 26(3):307-312. DOI:10.1016/j.difgeo.2007.11.022 · 0.59 Impact Factor
  • Toshiaki Adachi
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    ABSTRACT: In this paper, we study geodesics with null structure torsions on real hypersurfaces of type A 2 in a complex space form. These geodesics give a nice family of helices of order 3 generated by Killing vector fields on the ambient complex space form.
    Monatshefte für Mathematik 04/2008; 153(4):283-293. DOI:10.1007/s00605-008-0521-9 · 0.64 Impact Factor
  • Toshiaki Adachi, Sadahiro Maeda, Seiichi Udagawa
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    ABSTRACT: We give a necessary and sufficient condition for a Kähler manifold of complex dimension n ≧ 2 to be a complex space form in terms of its sectional curvatures, which is an extension of Schur’s lemma. Our study is related to a congruence theorem for circles in a complex space form
    Archiv der Mathematik 01/2008; 90(2):163-172. DOI:10.1007/s00013-007-2180-9 · 0.48 Impact Factor
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    SADAHIRO MAEDA, TOSHIAKI ADACHI
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    ABSTRACT: In this paper, we characterize hypersurfaces of type A2 in a complex projective space in terms of their geodesics.
    Bulletin of the Australian Mathematical Society 01/2008; 77(01):1 - 8. DOI:10.1017/S0004972708000014 · 0.48 Impact Factor
  • Sadahiro Maeda, Toshiaki Adachi
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    ABSTRACT: In a complex projective space, we distinguish hypersurfaces of type $({\rm A}_1)$ from hypersurfaces of type $({\rm A}_2)$ in terms of the cardinality of congruence classes of their extrinsic geodesics.
    Tohoku Mathematical Journal 01/2008; 60(2008). DOI:10.2748/tmj/1232376168 · 0.60 Impact Factor

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