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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
The subject of lectures is projective differential geometric treatment of surfaces and a relation with the hypergeometric differential equation.
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.
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.
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.
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.
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.
This is an overview on the relation of the theory of linear differential equations and the projective differential geometry.
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.
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.
This is an overview of Schwarzian derivatives that appear in several kinds of differential equations old and new; published as an article of Suugaku-no Tanoshimi 24(2001), 107--121.
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.
A detailed description on how to find the equation announced in the paper below.
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.
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.
This volume is the lecture notes on the subjects in the title at Dept Math, Brown Univ, 1988/89.
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.
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 including some of the author's biography; this is published as an article of Suugaku-no Tanoshimi 9(1998), 133-141.
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 Pl\"ucker embedding of the Grassmannians.
Abstract: 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.
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.
Abstract: 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.
Abstract: 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.
