Takeshi SASAKI

Deparment of Mathematics, Kobe University
Rokko, Kobe 657-8501, Japan

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Last updated on : September 30, 2015


Research Interests by keywords:


Recent (Pre)prints:

  1. An example of Schwarz map of reducible hypergeometric equation $E_2$ in two variables, coauthor: Keiji Matsumoto, Tomohide Terasoma and Massaki Yoshida, to appear in J. Math. Soc. Japan.

    We study an Appell's hypergeometric system $E_2$ of rank four which is reducible and we show that its Schwarz map admits geometric interpretations: the map can be considered as the universal Abel-Jacobi map of a 1-dimensional family of curves of genus 2.

  2. Reducibility of the systems of differential equations satisfied by Appell's $F_2$, $F_3$ and $F_4$, coauthor: K. Mimachi, Kyushu J. Math. 69(2015), 429--435.

    By using the contiguity operators, we give a simple and elementary derivation of sufficient conditions that Appell's hypergeometric systems for $F_2$, $F_3$ and $F_4$ are reducible.

  3. Schwarz maps for the hypergeometric differential equation, coauthor: S. Fujimori, M. Noro, K. Saji, and M. Yoshida, International J. Math. 26-5(2015), 1541002.

    We introduce the de Sitter Schwarz map for the hypergeometric differential equation as a variant of the classical Schwarz map. This map turns out to be the dual of the hyperbolic Schwarz map, and it unifies the various Schwarz maps studied before. an example is also studied.

  4. Projective minimality for centroaffine minimal surfaces, coauthor: Atsushi Fujioka, Hitoshi Furuhata, J. Geometry 105(2014), 87--102.

    We study projective minimality of centroaffine minimal surfaces and, by using that any centroaffine minimal surfaces have a one-parameter family of deformation known as associated surfaces, we shall give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are all projective minimal. We also show that any indefinite centroaffine minimal surface whose associated surfaces are all Godeaux-Rozet surfaces is a proper affine sphere.

  5. Monodromy representations associated with Appell's hypergeometric function F_1 using integrals of a multivalued function, coauthor: K. Mimachi, Kyushu J. Math. 66(2012), 89--114.

    Monodromy representations on the solution space of Appell's system of differential equations E_1 are studied by using integrals of a multivalued function. In particular, we realize the representations in the reducible cases and give a complete list of finite reducible representations.

  6. Irreducibility and reducibility of Lauricella's system of differential equations E_D and the Jordan-Pochhammer differential equation E_{JP}, coauthor: K. Mimachi, Kyushu J. Math. 66(2012), 61--87.

    Monodromy representations on the spacee of solutions of Lauricella's system of differential equations E_D and the Jordan-Pochhammer differential equation E_{JP} are studied by using integrals of a multivalued fundtion. We establish the fact that any solution of E_D and any solution of E_{JP} are both expressed by the integrals of a multivalued function. Then we give a necessary and sufficient condition for the monodromy representation to be irreducible.

  7. Monodromy representations associated with the Gauss hypergeometric function using integrals of a multivalued function, coauthor: K. Mimachi, Kyushu J. Math. 66(2012), 36--60.

    Monodromy representations on the spacee of solutions of the Gauss hypergeometric equation are studied by using integrals of a multivalued fundtion. We give a necessary and sufficient condition for the monodromy representation to be irreducible as well as a complete list of reducible representaions.

  8. The hypergeometric differential equation 3E2 with cubic curves as Schwarz images, coauthor: M. Kato, Kyushu J. Math. 65(2011), 55--74.

    We listed all cases where the Schwarz images associated with the hypergeometric differential equation 3E2 of rank 3 turn out to be cubic curves in the projective plane.

  9. Monodromy of the hypergeometric differential equation of type (3,6) III, (pdf) coauthor: K. Matsumoto and M. Yoshida, Kumamoto J. Math. 23(2010), 37--47.

    We found that, among two monodromy groups, where one is an arithmetic group acting on a symmetric domain and the other is the unitary reflection group that is sometimes denoted as ST34, a distinguished relation does hold.

  10. Discrete flat surfaces and linear Weingarten surfaces in hyperbolic 3-space, coauthor: T. Hoffman, W. Rossman, and M. Yoshida, Trans AMS 364(2012), 5605--5644.

    This paper proposes how to define discrete flat surfaces in hyperbolic 3-space by use of certain discrete integrable systems as well as to define discrete linear Weingarten surfaces.

  11. Asymptotic behavior of the hyperbolic Schwarz map at irregular singular points, coauthor: T. Koike and M. Yoshida, Funkcialaj Ekvacioj 53(2010), 99--132.

    In this paper, we study the asymptotic behavior of the hyperbolic Schwarz map defined in previous papers for the differential equation of order 2 with irreuglar singular points.

  12. Surface singularities appeared in the hyperbolic Schwarz map for the hypergeometric equation, (pdf) coauthor: Masaaki Yoshida, Progress in Math. 283(2009), 247--272.

    This is a survey on the hyperbolic Schwarz map associated with the hypergeometric differential equation presented in the Summer School on Arrangements, Local Systems and Singularities. This paper is dedicated to Professor Dr. Fritz Hirzebruch for his 80th birthday.

  13. Singularities of flat fronts and their caustics, and an example arising from the hyperbolic Schwarz map of a hypergeometric equation, coauthor: Masaaki Yoshida, Results in Math. 56(2009), 369--385.

    This describes in a detailed fashion how the caustics behave for the hyperbolic Schwarz map of a hypergeometric function. This paper is dedicated to the late Professor Katsumi Nomizu.

  14. Geometry of convex domains -- Method of affine differential geometry (in japanese; pdf) Jannuary, 2008.

    This is a survey on the geometry of convex domains in view of affine differential geometry, which treats the matters such as the characteristic function on convex domains, projectively invariant metrics, affine hyperspheres, differential equations associated with affine hyperspheres, area function of convex 2-domains, affine curvature flow of curves.

  15. Hyperbolic Schwarz maps of the Airy and the confluent hypergeometric differential equations and their asymptotic behaviors, coauthor: Masaaki Yoshida, J. Math. Sci. Univ. Tokyo 15(2008), 192--218.

    Hyperbolic Schwarz maps of the Airy differential equation and the confluent hypergeometric differential equation are examined. The change of behavior at infinity is investigated by use of asymptotic expansions of solutions. Among confluent hypergeometric differential equations with several parameters, Bessel differntial behaves very differntly from others. Several figures show these phenomena.

  16. Hyperbolic Schwarz map of the confluent hypergeometric differential equation, coauthor: Kentaro Saji and Masaaki Yoshida, J. Math. Soc. Japan 61(2009), 559--578.

    Hyperbolic Schwarz map of the confluent hypergeometric differential equation is studied from the aspect of singularities of the associated surfaces. The appearance of swallowtail singularities are fully presented and the types of singularities of the map are classified that depend on the parameters of the equation.

  17. Confluence of swallowtail singularites of the hyperbolic Schwarz map defined by the hypergeometric differential equation, coauthor: Masayuki Noro, Kotaro Yamada and Masaaki Yoshida, Experimental Math. 17(2008), 191--204.

    Flat surfaces in the three-dimesnional hyperbolic space have generically singularities. In one of previous papers, we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation. Its image is a flat front and its generic singularities are cuspidal edges and swallowtail singularities. In this paper we study the curves consisting of cuspidal edges and creation/elimination of swallowtail singularities depending on the parameters of the hypergeometric equation.

  18. Derived Schwarz map of the hypergeometric differential equation and a parallel family of flat fronts, coauthor: Kotaro Yamada and Masaaki Yoshida, Intern J. Math. 19(2008), 847--863.

    In a previous paper we defined a map, called the hyperbolic Schwarz map, from the one-dimensional projective space to the three-dimensional hyperbolic space by use of solutions of the hypergeometric differential equation, and thus obtained closed flat surfaces belonging to the class of flat fronts. We continue the study of such flat fronts in this paper. First, we introduce the notion of derived Schwarz maps of the hypergeometric differential equation and, second, we construct a parallel family of flat fronts connecting the classical Schwarz map and the derived Schwarz map.

  19. The hyperbolic Schwarz map for the hypergeometric differential equation, coauthor: Kotaro Yamada and Masaaki Yoshida, Experim Math. 17(2008), 269--282.

    The Schwarz map of the hypergeometric differential equation is studied since the beginning of the last century. Its target is the complex projective line, the 2-sphere. This paper introduces the hyperbolic Schwarz map, whose target is the hyperbolic 3-space. This map can be considered to be a lifting to the 3-space of the Schwarz map. This paper studies the singularities of this map, and visualize its image when the monodromy group is a finite group or a typical Fuchsian group. General cases will be treated in a forthcoming paper.

  20. Theory of surfaces and the hypergeomeric functions, (in japanese; pdf) March, 2006; Note of lectures given at the Early-Spring School on the hypergeometric functions.

    The subject of lectures is projective differential geometric treatment of surfaces and a relation with the hypergeometric differential equation.

  21. Interpolation of Markoff transformations on the Fricke surface, coauthor: Masaaki Yoshida, Tohoku Math. J. 60(2008), 23--36.

    We propose a way of interpolating the action of Markoff transformation on Fricke surfaces. As a particular result, we show that the space {(p,q,r); p^2+q^2+r^2-pqr-4=0, p>2, q>2, r>2} admits a GL(2,R)$-action extending the Markoff transformations.

  22. Homogeneous two-manifolds with an invariant two-form, coauthor: Masaaki Yoshida, J. Geometry 88(2008), 149--161.

    We study a 2-dimensional manifold that admits a homogeneous action of a 3-dimensional Lie group G, and has a 2-form invariant under G. We show that such a manifold can be realized as a surface in the affine 3-space, and list such realizations.

  23. Line congruence and transformation of projectve surfaces, Kyushu J. Math. 60(2006), 101--243.

    An overview on line congruences and transformations of surfaces in projective 3-space is given. It treats the subjects such as fundamentals on projective surfaces, line congruences, Laplace transformation of surfaces, linear complex, Weingarten congruences, Demoulin transformations, and projectively minimal surfaces.

  24. On the generalized Gauss hypergeometric system of three variables, coauthor: Katsushige Higashi, Kyushu J. Math. 58(2004), 105--120.

    The Pfaffian form of the generazlied Gauss hypergeometric system associated with Jack polynomials of three variables is presented and solutions around a singular point are given.

  25. Invariant theory for linear differential systems modeled after the Grassmannian Gr(n,2n), coauthor: Masaaki Yoshida, Nagoya J. Math. 171(2003), 163--186.

    Schwarzian derivative relative to a map from an n-manifold to the Grassmannian Gr(n,2n) is defined and its property is given. We apply the result to a 2-parameter family of line congruences.

  26. Schwarzian derivatives and uniformization, (pdf) coauthor: Masaaki Yoshida, CRM Proc. Lecture Notes, AMS 32(2002), 271--286.

    A survey on how Schwarzian derivatives associated with several kinds of differential systems are defined and on their applications to the uniformization problem related with certain moduli spaces.

  27. System of linear homogeneous differential equations and the projective differential geometry, (in japanese; pdf) Note of a lecutre given at 50-th Differential Geometry Symposium in 2003, "21Seiki no Suugaku -- Kikagaku no Mitouhou", Nihon hyoronsha, 2004, pp. 281--291.

    This is an overview on the relation of the theory of linear differential equations and the projective differential geometry.

  28. The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces II, coauthor: Masaaki Yoshida, J. Phys. A:Math Gen 34(2001), 2319--2328.

    Continuing the paper with the same title (of part I), we showed how to obtain the uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cuibc surfaces.

  29. A geometric study of the hypergeometric function with imaginary exponents, coauthor: Masaaki Yoshida, Experimental Math. 10(2001), 321--330.

    The behavior of the map associated with Gauss hypergeometric function highly depends on the value of exponents. When exponents are imaginary, we encounter a completely different picture compared with the case when exponents are real. Several pictures are exhibited.

  30. Around the Schwarzian derivatives, (in japanese; pdf) Suugaku-no Tanoshimi 24(2001), 107--121.

    This is an overview of Schwarzian derivatives that appear in several kinds of differential equations old and new.

  31. Projective surfaces defined by Appell's hypergeometric systems E4 and E2, Kyushu J. Math. 55(2001), 1--21.

    The two systems in the title define surfaces in the projective 3-space. According to the value of parameters, the surfaces have different geometric properties. While the system E4 gives a 3-parameter family of projectively applicable surfaces, we examine the cases where the surfaces are cuibc or projectively minimal surfaces for the above two systems.

  32. A system of differential equations in $4$ variables of rank 5 invariant under the Weyl group of type E6, (pdf) coauthor: Masaaki Yoshida, Kobe J. Math. 17(2000), 29--57.

    A detailed description on how to find the equation announced in the paper below.

  33. The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces, coauthor: Masaaki Yoshida; Proc. Japan Acad. 75(1999), 129--133.

    In this paper, we found the uniformizing differential equation that governs the developing map of a complex hyperbolic structure on the 4-dimensional moduli space of marked cubic surfaces. Our equation is invariant under the action of the Weyl group of type E6.

  34. Globally defined linear connections on the real line and the circle, coauthor: Katsumi Nomizu; Tohoku Math. J., 51(1999), 205-212.

    We classified globally defined linear connections on the real line as well as on the circle up to diffeomorphisms and proved that such connections can be realized by affine immersions into the affine plane.

  35. Projective Differential Geometry and Linear Homogeneous Differential Equations, (pdf) Rokko Lectures in Mathematics vol 5, 1999, pp. 114;

    This volume is the lecture notes on the subjects in the title at Dept Math, Brown Univ, 1988/89.

  36. The Classification of projectively homogeneous surfaces II, coauthors: Franki Dillen and Luc Vrancken, Osaka J. Math. 35(1998), 117-146.

    This paper shows how to classify projectively homogeneous surfaces in projective 3-space; the list of such surfaces complements the list of projectively homogeneous surfaces with non-vanishing Fubini-Pick invariant previously given by K. Nomizu and T. Sasaki, and thus completing the classification. The method is to construct projective invariants for such surfaces; this paper belongs to projective differential geometry in terms of affine connection and projective immersion.

  37. G. Darboux: Theorie Generale des Surface, (in japanese; pdf) Suugaku-no Tanoshimi 9(1998), 133-141.

    This is a review of the old volumes by G. Darboux with the title Theorie Generale des Surface from the reviewer's personal viewpoint emphasized on the transformation of surfaces.

  38. On the rigidity of differential systems modelled on hermitian symmetric spaces and disproofs of a conjecture concerning modular interpretations of configuration spaces, (pdf) coauthors: K. Yamaguchi and M. Yoshida, Advanced Studies in Pure Mathematics 25, 1997, "CR-Geometry and Overdetermined Systems," pp. 318--354.

    The equivalence of differential systems associated with semi-simple Lie algebras can be encoded in certain cohomology groups. We study the cohomology for the differential systems modelled on hermitian symmetric spaces and we apply it to the proof of the nonequivalence of the hypergeometric system E(k,n) and the system of Pluecker embedding of the Grassmannians.

  39. Sectional curvature of projective invariant metrics on a strictly convex domain, coauthor:T.Yagi, Tokyo Journal of Mathematics, 19, 1996, 419--433.

    The sectional curvature of certain naturally defined invariant metrics on a strictly convex domain is computed and it is shown that the curvature tends to -1 at the boundary.

  40. On the system of differential equations associated with a quadric and hyperplanes, coauthored with K.Matsumoto, Kyushu J. Math. 50(1996), 93--131.

    Study of hypergeometric integrals associated with the configuration of one quadratic hypersurface and any number of hyperplanes in the projective space. Such a configuration includes that of Appell's F4 and a 5-dimensional family of K3 surfaces.

  41. Inflection points and affine vertices of closed curves on 2-dimensional affine flat tori, Results in Math., 27(1995), 129--140.

    This paper deals with a geometric problem on inflection points and affine vertices for closed curves in an affine flat torus. We show that the least number of inflection points lying on a closed curve that is not homotopic to zero is 2 if the torus is affinely equivalent to a euclidean torus and 0 otherwise. We consider also the number of affine vertices on a strictly convex closed curve on a flat torus. An explicit example of a closed curve with six affine vertices is given.

  42. Affine immersion of $n$-dimensional manifold into $R^{n+n(n+1)/2}$ and affine minimality, Geometriae Dedicata 57(1995), 317--333.

    We formulate an affine theory of immersions of an $n$-dimensional manifold into the euclidean space of dimension $n+n(n+1)/2$ and give a characterization of critical immersions relative to the induced volume functional in terms of the affine shape operator.


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