Bonnet Surfaces and Isothermic Surfaces

By the study of the relations between Bonnet surfaces and isothermic surfaces, we obtain classification results of Bonnet surfaces in 3-dimensional space form R³(c) and of the spacelike Bonnet surfaces in indefinite space form R₁ ³(c), which generalize the results in Bobenko’s [1] and Peng-Lu’s [11]. It is remarkable that there exist always Bonnet surfaces which are not Weingarten surfaces, if the ambient space is not R³(c) for c ≥ 0.     

Some Global Theorems on Closed Surfaces With Qenus Zero in E~3

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Lagrangian Bonnet Problems in Complex Space Forms

In this note we consider Lagrangian Bonnet problem for Lagrangian surfaces in complex space forms. We first give a Bonnet type theorem for conformal Lagran     

Global results on Bonnet Surfaces

The local theory of the Bonnet Surfaces in the three dimensional Euclidean Space of the type of non-constant mean curvature that admit infinitely many non-trivial and geometrically distinct isometries preserving the mean curvature has been developed, in the literature, under the assumption that the surfaces contain no umbilic points and no critical points of the mean curvature function. Here, we prove that these restrictions do not create any difference from the possible global results, except in one case in which we prove that the set of the umbilic points, known to consist of exactly one point, is equal to the set of the critical points of the mean curvature function. Furthermore, we show that the index of this umbilic point, as isolated singularity of the foliation of the principal curves is one. In our proofs we use: (a) An intrinsic characterization of these surfaces, which we derive in a manner easier and including more details than those already found in the literature. From this characterization we conclude that all surfaces of this type are analytic. (b) The harmonic functions of the angles by which the respected isometries rotate the principal frames, which we compute.Dedicated to the memory of my great benefactors: my first important mathematics teacher and motivator, my uncle Michael Ioannou Roussos ( November, 1992), and his wife, my aunt Evanggelia Michael Roussou-Gavala ( May, 1994), for their invaluable support.     

Reductions of Gauss–Codazzi Equations

We prove that conformally parameterized surfaces in Euclidean space of curvature c admit a symmetry reduction of their Gauss–Codazzi equations whose general solution is expressed with the sixth Painlevé function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko et al. 1997) can be recovered by such a reduction.     

R_1~3中混合型曲面保主曲率等距变形

本文研究了Minkowski空间R_(1)~3曲面的等距变形问题.建立了R_(1)~3中曲面的共形、等距等概念.推广了O.Bonnet和S.S.Chern关于欧氏空间的结论.对R_(1)~3出现的新情况——曲面的中曲率梯度类光作了一定探讨,得出的主要结果为:非平坦的、允许保主曲率等距变形的曲面一定不是W-曲面.     

R4(C)中平均曲率方向平行的Bonnet曲面

本文利用[5]的方法,对R4(C)中平均曲率方向平行的Bonnet曲面引入了半测地等温 参数,并给出了一个分类结果.     

Bonnet Ruled Surfaces

We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.     

R4(C)中平均曲率方向平行的Bonnet曲面

本文利用[5]的方法,对R⁴(C)中平均曲率方向平行的Bonnet曲面引入了半测地等温 参数,并给出了一个分类结果.     

Two-dimensional almost-Riemannian structures with tangency points

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss–Bonnet formula for almost-Riemannian structures with tangency points.     

Coincidence Wecken homotopies versus Wecken homotopies relative to a fixed homotopy in one of the maps

We study the 1-parameter Wecken problem versus the restricted Wecken problem, for coincidence free pairs of maps between surfaces. For this we use properties of the function space between two surfaces and of the pure braid group on two strings of a surface. When the target surface is either the 2-sphere or the torus it is known that the two problems are the same. We classify most pairs of homotopy classes of maps according to the answer of the two problems are either the same or different when the target is either projective space or the Klein bottle. Some partial results are given for surfaces of negative Euler characteristic.     

Der Satz von Bonnet für (k+1)-Regelflächen

In the theory of ruled surfaces a theorem of O. Bonnet is well-known which is concerned with the striction line and its properties to be a geodesic and an isogonal trajectory. In this paper we will investigate (k+1)-dimensional generalized ruled surfaces generated by a one-parameter family ofk-dimensional linear subspaces of then-dimensional Euclidean spaceE _n . A theorem will be given which is the best generalisation of the result of Bonnet for generalized ruled surfaces.     

Global geometry and topology of spacelike stationary surfaces in the 4-dimensional Lorentz space

For spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space, one can naturally introduce two Gauss maps and a Weierstrass-type representation. In this paper we investigate the global geometry of such surfaces systematically. The total Gaussian curvature is related with the surface topology as well as the indices of the so-called good singular ends by a Gauss–Bonnet type formula. On the other hand, as shown by a family of counterexamples to Osserman?s theorem, finite total curvature no longer implies that Gauss maps extend to the ends. Interesting examples include the deformations of the classical catenoid, the helicoid, the Enneper surface, and Jorge–Meeks? k-noids. Each family of these generalizations includes embedded examples in the 4-dimensional Lorentz space, showing a sharp contrast with the 3-dimensional case.     

A^3中曲面的无穷小等距变形

本文讨论了三维仿射空间Ａ＾３中曲面的无穷小等距变形,给出了Ａ＾３中无穷小Ｏ．Ｂｏｎｎｅｔ等距的定义,得到了曲面允许非平凡的无穷小Ｏ．Ｂｏｎｎｅｔ等距变形的一个充要条件。     

Boundary value problems for quaternionic monogenic functions on non-smooth sureaces

In this paper, analogous of the Compound Riemann-Hilbert boundary value problems are investigate for quaternionic monogenic functions. The solution (explicitly) of the problem is established over continuous surface, with little smoothness, which bounds a bounded domain of R³. In particular, smoothness property for high-dimensional Cauchy type integral are computed. We also use Zygmund type estimates to adapt existing one-variable complex results to ilustrate the Hölder-boundedness of the singular integral operator on 2-dimensional Ahlfors regular surfaces. At the end, uniqueness of solution for the Riemann boundary value problem have already built taking as a base the general Operator Theory.     

On the problem of isometry of a hypersurface preserving mean curvature

The problem of determining the Bonnet hypersurfaces in R ⁿ⁺¹, for n > 1, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called Bonnet associate of the initial hypersurface. The orthogonal net which is called A-net is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in R ⁿ⁺¹ with no umbilical points is a Bonnet hypersurface if and only if it has an A-net.     

On index formulas for manifolds with metric horns

In this paper was discuss the index problem for geometric differential operators (Spin–Dirac opreatorm Gauβ–Bonnet operator, Singnature operator) on manifolds with metric horns. On singular manifolds these operators in general do not unique closed extensions. But there always exist two extremal extensions D$sub:min$esub:and D$sub:max$esub: We describe the quotient D (D$sub:max$esub:)/D(D$sub:esub:) explicitely in geometric resp.topological terms of the base manifolds of the metric horns.

We derive index formulas for the3 Spin–Dirac and GauβBonnet operator. For the Singnature operator we present a partial result.     

Bonnet Surfaces with Constant Curvature

We classify Bonnet surfaces with constant curvature in 3-dimensional space forms and show that they are parametrized by curves in 2-dimensional space forms with specific geodesic curvature.     

Analytical solution of the quasi-static thermoelasticity problem in a pressurized thick-walled cylinder subjected to transient thermal loading

Thermoelasticity problem in a thick-walled cylinder is solved analytically using the finite Hankel transform. Time-dependent thermal boundary conditions are assumed to act on the inner surface of the cylinder. For the mechanical boundary conditions two different cases are assumed: Traction–displacement problem (traction is prescribed on the inner surface and the fixed displacement boundary condition on the outer one) and Traction–Traction problem (tractions are prescribed on both the inner and outer surfaces of the hollow cylinder). The quasi-static solution of the thermoelasticity problem is derived analytically, i.e., the transient thermal response of the cylinder is derived and then, quasi-static structural problem is solved and closed form relations are extracted for the thermal stresses in the two problems. The results show to be in accordance with that cited in the literature in the special cases.     

Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving a suitable initial value problem for the corresponding elastic curves. The behavior of such elastic curves is examined by a method provided by Langer and Singer to reduce the order of the underlying ordinary differential equation. This ensures that these solutions differ from solutions already obtained by Dall’Acqua, Deckelnick and Grunau. We will additionally provide a Bernstein-type result concerning the profile curve of a Willmore surface of revolution. If this curve is a graph on the whole real numbers it has to be a Möbius transformed catenary. We show this by a corollary of the above-mentioned method by Langer and Singer.