Gauss Image Research Articles

The asymptotic blow-up of a surface in Euclidean 3-space

Given a smoothly immersed surface in Euclidean (or affine) 3-space, the asymptotic directions define a subset in the Grassmann bundle of unoriented one-dimensional subspaces over the surface. This links the Euler characteristic of the region where the Gauss curvature is nonpositive with the index of singularities in a natural line field defined on this subset. To apply this we need only identify mechanisms which restrict the index of the singularities. In Section 2.1 we show that specific configurations of nonpositive Gauss curvature cannot be realized by an immersed surface and that specific configurations in 2-sphere cannot be realized as Gauss images of surfaces. In Section 2.2 we prove an existence theorem for surfaces which satisfy regularity conditions and a Symplectic Monge Ampere PDE. In general, a PDE of this type will not restrict the indices of the singularities over a solution. However, we show that over a surface of nonzero constant mean curvature the indices are restricted and, hence, that specific configurations of nonpositive Gauss curvature cannot be realized by a constant mean curvature surface.

ガウス写像による工学的曲面分類とその応用

In the computer-aided designs for castings or forgings, we encounter many complex shaped surfaces, which are difficult to create or evaluate even with use of modern CAD systems. Further, structural or mechanical engineers are annoyed with lack of methodology to evaluate adaptiveness of generated surfaces; conventional graphic tools are unuseful in the above cases.In the present study, one proposes an engineering method to deal with curved surfaces appearing in the practical designs. Based on the differential geometry and computational processings, the proposed approach employs an indicator for evaluation of curved surfaces, and gauss image is applied to represent the distribution of gauss curvetures. Through rough comparison of generated gauss images, one can see differences of curved surfaces. Furthermore, using the present approach, one can estimate the graphical accuracy of approximate triangular development method for both developable and undevelopable surfaces.

Global properties of the Gauss image of flat surfaces in ${\bf R}^4$

Global properties of the Gauss image of flat surfaces in ${\bf R}^4$

The Gauss image of flat surfaces in ${\bf R}^4$

The Gauss image of flat surfaces in ${\bf R}^4$

Experimental Study of the Laser Speckle Phase in the Image Field

Statistical properties of the laser speckle phase in the image field were experimentally studied under the condition that the speckle field obeys Gaussian statistics. The speckle phase was measured in relation to the phase variations of light passing through a diffuse object and the defocusing of the imaging system. It is found that the phase variations of speckles at the image plane are smaller than those of light passing through the object. The anomaly of the speckle phase in the neighbourhood of the Gauss image plane is also found and interpreted in relation to the dip phenomenon of the speckle contrast.

O(3) nonlinear σ-model and pseudospherical surface

The authors relate the dual symmetry of the chiral field to the dual symmetry of the first and second forms on pseudospherical surfaces in asymptotic coordinates. Every given (up to conformal similarity) solution of chiral field on O(3)/O(2) sphere has been put into one-to-one correspondence with the Gauss image of a definite (up to homothetic transformations) pseudospherical surface. They establish a gauge covariant formulation which unifies the chiral field and the differential forms on the pseudospherical surface. Then they get the explicit geometrical pictures and the covariant relations for Backlund transformations, Riccati equations and an infinite number of conserved currents in both cases. The SO(3) and SO(2,1) sigma -model are also related by dual symmetry.

Minimal surfaces in R5 whose Gauss images have constant curvature

Minimal surfaces in R5 whose Gauss images have constant curvature

Submanifolds with totally geodesic Gauss image

Submanifolds with totally geodesic Gauss image

Deformation of a submanifold in a Euclidean space with fixed gauss image: II

Deformation of an m-dimensional submanifold immersed in a Euclidean n-space is studied when the Gauss image does not move pointwise. The present paper is the third one and contains deformations where the tensor of deformation satisfies some differential equations, normal deformations and deformations of totally-real submanifolds in a complex space.

Deformation of a submanifold in an Euclidean space with fixed Gauss image

Deformation of a submanifold in an Euclidean space with fixed Gauss image

Deformability of a submanifold in a Euclidean space whose image by the Gauss map is fixed

In the present paper, deformations of an immersion of an m-dimensional manifold M in Euclidean n-space R n {R^n} such that the Gauss image is fixed pointwise are studied. We call such deformations admissible deformations. The main theorem gives a necessary and sufficient condition for an immersed manifold to admit nontrivial admissible deformations.

Immersionen mit paralleler zweiter Fundamentalform: Beispiele und Nicht-Beispiele

1. We show that symmetric R-spaces can. be imbedded in euclidean space with parallel second fundamental tensor (Dα=0). 2. We give a restrictive necessary condition for totally geodesic submanifolds of the Grassmannian to be the Gauss image of an immersion with Dα=0, c.f. [9].

Berstein Type Theroms Without Graphic Condition

Abstract. We prove Bernstein type theorems for minimal n-submanifolds in R n+p with flatnormal bundle under certain growth conditions for p ≥ 2 and n ≤ 5, as well as for arbitrary n andp = 1. When M is a graphic minimal hypersurface we recover the known result.Key words. minimal submanifold, flat normal bundleAMS subject classifications. 49Q05, 53A07, 53A10 1. Introduction. There are various generalizations of the Bernstein theorem .For an entire minimal graph M = (x,f(x)) of dimension n ≤ 7 in R n+1 the problemhas been settled in [Si], [B-G-G]. If there is no dimension limitation, we have resultsin [M], [C-N-S], [Ni] and [E-H] under the growth condition on the function f.For general minimal hypersurfaces, a natural condition is stable, or even oneswith finite index. In this situation [C-S-Z] and [L-W] conclude that they have onlyone end or finitely many ends, respectively.Higher codimensional Bernstein problem becomes more complicated. In [H-J-W] Moser’s result has been generalized to minimal graphs of higher codimension.Recently, we obtained better results under a bound for the slope of the vector-valuedfunctions f which is independent of the dimension and codimension [J-X1] [J-X2](alsosee [Wm]. On the other hand, the counter example of [L-O] prevents us from goingfurther.For general minimal surfaces in a Euclidean space, so-called parametric case, theBernstein theorem was generalized to the beautiful theory of the value distribution ofits Gauss image [Os] [Xa] [F].In the present article we study minimal n-submanifolds of R