This volume presents the cutting-edge contributions to the Seventh International Workshop on Complex Structures and Vector Fields, which was organized as a continuation of the high successful preceding workshops on similar research.
The volume includes works treating ambitious topics in differential geometry, mathematical physics and technology such as Bézier curves in space forms, potential and catastrophy of a soap film, computer-assisted studies of logistic maps, and robotics.
https://doi.org/10.1142/9789812701763_fmatter
PREFACE
CONTENTS
CONTRIBUTED COMMUNICATIONS
https://doi.org/10.1142/9789812701763_0001
We give a survey on constructing real surfaces associated with trajectories for Kähler magnetic fields and on comparing sectors and crescents on these surfaces.
https://doi.org/10.1142/9789812701763_0002
In this paper we give a survey on ordinary helices which are integral curves of Killing vector fields on symmetric spaces of rank one. On a real space form ℝn, Sn or Hn, all ordinary helices are generated by some Killing vector fields, and they are congruent each other if they have the same curvatures. But the situation is not the same for other symmetric spaces of rank one. Even a complex hyperbolic space admits bounded ordinary helices. We also make mention of an example of closed ordinary helices in a complex projective plane with 6 self-intersection points.
https://doi.org/10.1142/9789812701763_0003
The result in [KL] and a canonical construction will be used to obtain the real analyticity of the almost Kähler manifolds.
https://doi.org/10.1142/9789812701763_0004
In this paper, we characterize parallel immersions of rank one Riemannian symmetric spaces into a real space form by using the notion of isotropic immersions and inequalities related to mean curvatures.
https://doi.org/10.1142/9789812701763_0005
In this paper the interconnection between holomorphic and analytic functions is studied in the case of the algebra of bi-complex numbers. It is proved that each holomorphic bi-complex function is bi-complex analytic. In the general case of associative complex algebras this is not true.
https://doi.org/10.1142/9789812701763_0006
This paper considers a generalization of the existing concept of parallel (with respect to a given connection) geometric objects and its possible usage as a suggesting rule in searching for adequate field equations and local conservation laws in theoretical physics. The generalization tries to represent mathematically the two-sided (or dual) nature of the physical objects, the change and the conservation. The physical objects are presented mathematically by sections Ψ of vector bundles, the admissible changes DΨ are described as a result of the action of appropriate differential operators D on these sections, and the conservation proprieties are accounted for by the requirement that suitable projections of DΨ on Ψ and on other appropriate sections must be zero. It is shown that the most important equations of theoretical physics obey this rule. Extended forms of Maxwell and Yang-Mills equations are also considered.
https://doi.org/10.1142/9789812701763_0007
We consider the action of SO(N, C)-action on simply connected minimal surfaces in the N-dimensional Euclidean space RN. If the minimal surface is holomorphic with respect to an appropriate orthogonal complex structure on RN, then SO(N, C) preserves the holomorphicity. In this paper, we shall study minimal surfaces appering as bubbles under the action of SO(N, C) and prove that if SO(N, C) preserves the stableness of the minimal surface, then the minimal surface is holomorphic with respect to an orthogonal complex structure on RN.
https://doi.org/10.1142/9789812701763_0008
In this paper we formulate a Laplace-transform multiple scale expansion procedure to develop asymptotic solution of weakly non-linear partial differential equation. The method is applied to some general nonlinear wave and diffusion equations.
https://doi.org/10.1142/9789812701763_0009
Special compositions in an n-dimensional Weyl space are studied in [6], [7], [10], [11] and [2]. Compositions, generated by nets in an n-dimensional Weyl space are introduced in [12] and [13]. This paper is devoted to the study of special orthogonal compositions in a four-dimensional Weyl space. In the second paragraph there are found geometry characteristics of orthogonal Cartesian and quasichebyshevian compositions about the Weyl connection and about the Levi-Civita connection. The form of space, containing these compositions, is defined. In the third paragraph there are given the curvature properties on a four-dimensional Weyl space, containing special orthogonal compositions. There are found invariant tensors and compositions about a conformal transformation on a Weyl space.
https://doi.org/10.1142/9789812701763_0010
Some elements of classical mechanics and classical statistical mechanics are formulated in terms of fibre bundles. In the bundle approach the dynamical and distribution functions are replaced by liftings of paths in a suitably chosen bundle. Their time evolution is described by appropriate linear transports along paths in it or, equivalently, by corresponding invariant bundle equations of motion. In particular, the bundle version of the Liouville equation is derived.
https://doi.org/10.1142/9789812701763_0011
In the present note, some recent results concerning the existence of indefinite Kähler metrics of constant scalar curvature (e.g., indefinite Kähler-Einstein metrics and scalar-flat Kähler metrics) on compact complex surfaces are reported. It turns out that such an existence problem is closely related to a certain obstruction, which is a generalized version of the Bando-Calabi-Futaki character. Related problems and questions are also proposed.
https://doi.org/10.1142/9789812701763_0012
We give an announcement of our work on drawing Bézier curves on a real space form. By extending the notion of rational Bézier curves on a Euclidean plane, we define projective Bézier curves. We show we can draw circle-arcs by them.
https://doi.org/10.1142/9789812701763_0013
The algebraic notion of Gieseker stability is related to the existence of balanced metrics which are zeros of a certain moment map. We investigate some properties of balanced metrics relative to the Harder-Narasimhan filtration of a vector bundle and to blowups in the case of projective surfaces.
https://doi.org/10.1142/9789812701763_0014
The main purpose of this paper is to survey characterizations of totally umbilic hypersurfaces and isoparametric hypersurfaces related to the results in [1] and [3].
https://doi.org/10.1142/9789812701763_0015
In a recent paper two of us (J.Ł. and L.M.T.S., 2001) have given a complete list (modulo the (8,8)-periodicity) of 145 basic type-changing transformations of Hurwitz pairs. By the Complementarity Theorem (Thm. 3 of that paper) it is enough to study Hermitian Hurwitz pairs (responsible for 49 such transformations) which, by the Atomization Theorem due to J.Ł. and O.Suzuki (2001) connects quasiregular functions in the sense of Clifford analysis with hyperkählerian holomorphic chains (P. Dolbeault, J. Kalina, and J.Ł., 1999). Explicitly, using fractal representation of Clifford algebras corresponding to Hermitian Hurwitz pairs we are able to prove a counterpart of the Atomization Theorem for fractal gemmae, distinguishing 11 basic type-changing transformations corresponding to 11 atoms appearing in the original Atomization Theorem.
https://doi.org/10.1142/9789812701763_0016
In this paper, a lot of examples of four-dimensional manifolds with an almost hypercomplex pseudo-Hermitian structure are constructed in several explicit ways. The received 4-manifolds are characterized by their linear invariants in the known aspects.
https://doi.org/10.1142/9789812701763_0017
The Navier-Stokes equations are considered by the use of the method of Lagrangians with covariant derivatives (MLCD) over spaces with affine connections and metrics. It is shown that the Euler-Lagrange equations appear as sufficient conditions for the existence of solutions of the Navier-Stokes equations over (pseudo) Euclidean and (pseudo) Riemannian spaces without torsion. By means of the corresponding (n − 1) + 1 projective formalism the Navier-Stokes equations for radial and tangential accelerations are found.
https://doi.org/10.1142/9789812701763_0018
Our purpose of this paper is to show how the quaternion formalism can be applied with great success not only to the interpolation between coordinate frames, but also to a remarkably elegant description of the evolving coordinate-frame geometry of curves, Specific applications of these techniques include the generation of optimal kimematics solution corresponding to smooth mathematical curves appearing in computer graphics or scientific molecular kinematics. The correspondence between the orientation of a 3D object represent by a 3 × 3 orthogonal matrix in the group SO(3) and unit quaternions has long known to physicists and mathematicians, and was brought to attention of the computer graphics community by Shoemake, 1985. Unit quaternions are isomorphic to the topological 3-sphere S3, which is also the topological space of the Lie group SU(2), the simply connected twofold cover of the group SO(3) describing rotations in ordinary 3D Euclidean space.
https://doi.org/10.1142/9789812701763_0019
In the present note, we mainly review existence theorems of the indefinite metrics of signature (+ + − −), i.e., neutral metrics in four dimension, and the existence of two kinds of almost complex structures which are associated with such neutral metrics. Recent results on Walker 4-manifolds, which exhibit a large variety of neutral geometries, are also included.
https://doi.org/10.1142/9789812701763_0020
Asymptotics of O.D.E. appearing in the turning point problems can be characterized literally by its characteristic polygon. The simplest case is the Airy equation which has a one-segment characteristic polygon. Nakano [15], [16], [20]~[22], Nakano et al. [23], and Roos [28], [29] study O.D.E.’s with a several-segement one. Here, we study an O.D.E. with an arbitrarily finitely many-segment one.
https://doi.org/10.1142/9789812701763_0021
We calculate derivatives of projective Bézier curves of order 2 and study the C2 condition in joining them.
https://doi.org/10.1142/9789812701763_0022
We search for the shape of a soap film stretched to a certain boundary condition by numerical computation. When the shape for one boundary condition is not only one, we consider which shape is a true soap film among these. We also consider about the phenomenon which a soap film cause when we change this boundary condition.
https://doi.org/10.1142/9789812701763_0023
The following note explains, how one may extend “kinematics in number Spaces”, (especially from [B2], [Fr/Sp], [Sp7]) to kinematics in differentiable Spaces, i.e. in “manifolds with or without singularities”, and with - in singularity-cases - relevant locally integrable vectorfields ([Sp1],[Sp2]). Singularities appear already in classical situations and hence should be included in the theory right from the beginning. This needs some new and more general start than in [Fr/Sp]. From the starting points in this note one may try to go on along similar lines as in [Fr/Sp]. Special cases are known from classical branches of kinematics: See the literature in [P] for more classical examples, see especially [B2], [Fr/Sp], [Sp7] for some later development. Early beginnings of this general frame in some special case are due especially to H.R. Müller [M] and O. Giering, H. Frank [F/G], [F]. As an extension of [Fr/Sp], this paper is written in german.
https://doi.org/10.1142/9789812701763_0024
No abstract received.
https://doi.org/10.1142/9789812701763_0025
No abstract received.
https://doi.org/10.1142/9789812701763_0026
The class of the complex manifolds with Norden metric is considered. The Yano connection is introduced. The properties of the curvature tensor and the Bochner tensor of the Yano connection are studied.
https://doi.org/10.1142/9789812701763_0027
In this article, we introduce two topics of our recent works ([4], [6], [8]). One is about the integrability of almost hyperhermitian manifolds. The other is about 4-dimensional almost hyperhermitian manifolds, especially.