Non-degenerate Surfaces Research Articles

Flat Affine Surfaces in $$\mathbb{R}^4 $$ with Flat Normal Connection

In this paper we study nondegenerate affine surfaces in the 4-dimensional affine space \(\mathbb{R}^4 \). We assume that both the connection ∇ and the normal connection ∇⊥ induced by the canonical equiaffine transversal bundle are flat. Surfaces with constant equiaffine transversal bundle are trivial examples of such surfaces. Here, we obtain a complete classification of all such surfaces which do not have constant equiaffine normal bundle.

The discrete Plateau Problem: Algorithm and numerics

We solve the problem of finding and justifying an optimal fully discrete finite element procedure for approximating minimal, including unstable, surfaces. In this paper we introduce the general framework and some preliminary estimates, develop the algorithm, and give the numerical results. In a subsequent paper we prove the convergence estimate. The algorithmic procedure is to find stationary points for the Dirichlet energy within the class of discrete harmonic maps from the discrete unit disc such that the boundary nodes are constrained to lie on a prescribed boundary curve. An integral normalisation condition is imposed, corresponding to the usual three point condition. Optimal convergence results are demonstrated numerically and theoretically for nondegenerate minimal surfaces, and the necessity for nondegeneracy is shown numerically.

On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of R 3 {\mathbf R}^3 by nontrivial translations or by screw motions with nontrivial rotational part, fixed finite topology and planar type ends, are endowed with natural structures of finite dimensional real analytic manifolds. This nondegeneracy is defined in terms of Jacobi functions. Riemann’s minimal examples are characterized as the only nondegenerate surfaces with genus one in their corresponding spaces. We also give natural immersions of these spaces into certain complex Euclidean spaces which turn out to be Lagrangian immersions with respect to the standard symplectic structures.

Classification of flat indefinite Equicentroaffinely Homogeneous Surfaces in R4

A nondegenerate equi-centroaffine surface in R4 is called homogeneous if for any two points p and q on the surface there exists an equi-centroaffine transformation in R4 which takes the surface to itself and takes p to q. In this paper we classify the equi-centroaffinely homogeneous surfaces with flat indefinite metric in R4 up to centroaffine transformations.

Affine Dupin Surfaces

In this paper we study nondegenerate affine surfaces in R 3 \mathbb {R} ^{3} whose affine principal curvatures are constant along their lines of curvature. We give a complete local classification of these surfaces assuming that the lines of curvature are planar, and there are no umbilics.

On rigidity of affine surfaces

Rigidity of nondegenerate Blaschke surfaces in R 3 \mathbf {R}^{3} is studied. The rigidity criteria are given in terms of ∇ R \nabla R , where R R is the curvature of the Blaschke connection ∇ \nabla . If the rank of ∇ R \nabla R is 2, then the surface is rigid. If ∇ R = 0 \nabla R=0 , it is nonrigid. In the case where the rank of ∇ R \nabla R is 1 there are both rigid and nonrigid surfaces. This case is discussed for various types of surfaces.

Affine Surfaces whose Geodesics are Planar Curves

We study the geometry of nondegenerate affine surfaces M2 in R4 , with respect to the Burstin-Mayer, the Weise-Klingenberg and the equiaffine transversal plane bundle. A classification is obtained of the surfaces whose geodesies with respect to the induced connection are planar curves.

Affine surfaces whose geodesics are planar curves

We study the geometry of nondegenerate affine surfaces ${M^2}$ in ${\mathbb {R}^4}$, with respect to the Burstin-Mayer, the Weise-Klingenberg and the equiaffine transversal plane bundle. A classification is obtained of the surfaces whose geodesies with respect to the induced connection are planar curves.

An affine characterization of the Veronese surface

IfM2 is a nondegenerate surface in a 4-dimensional Riemannian manifold\(\tilde M\), then there is a natural affine metricg defined onM2. It is shown that this affine metricg is conformal to the induced Riemannian metric onM2 if and only ifM2 is a minimal submanifold of\(\tilde M\) in the usual Riemannian sense. If the conformal factor is a constant, then the two metrics are said to be homothetic. It is shown that there does not exist a nondegenerate surface in Euclidean space ℝ4 or hyperbolic spaceH4 whose affine metric is homothetic to the induced Riemannian metric. Furthermore, ifM2 is a nondegenerate surface in the standard 4-sphereS4 whose affine metric is homothetic to the induced Riemannian metric, thenM2 is a Veronese surface.

Surfaces in ℝ 4 with Constant Affine Gauss Maps

In this paper we study the affine Gauss maps of nondegenerate surfaces in R4 with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R4 with constant affine Gauss maps.

Surfaces in 𝑅⁴ with constant affine Gauss maps

In this paper we study the affine Gauss maps of nondegenerate surfaces in R 4 {\mathbb {R}^4} with respect to the Burstin-Mayer normalization (1927) and with respect to the normalization obtained by Nomizu and the first author in 1993. We determine up to affine transformations all surfaces in R 4 {\mathbb {R}^4} with constant affine Gauss maps.

A NEW EQUIAFFINE THEORY FOR SURFACES IN ℝ4

In this paper, we investigate the geometry of nondegenerate affine surfaces in ℝ4. The main idea is to introduce an equiaffine structure on the surface by constructing a canonical transversal plane field with the aid of the affine metric. As an application, we then investigate surfaces with vanishing cubic forms. If the affine metric is positive-definite, such a surface can be locally described as a complex curve in [Formula: see text]. On the other hand, if the affine metric is indefinite, such a surface can be seen as the product of two planar curves.

Cohomology of the symplectic group 𝑆𝑝₄(𝑍). I. The odd torsion case

Let h 2 {h_2} be the degree two Siegel space and S p ( 4 , Z ) Sp(4,\mathbb {Z}) the symplectic group. The quotient S p ( 4 , Z ) ∖ h 2 Sp(4,\mathbb {Z})\backslash {h_2} can be interpreted as the moduli space of stable Riemann surfaces of genus 2 2 . This moduli space can be decomposed into two pieces corresponding to the moduli of degenerate and nondegenerate surfaces of genus 2 2 . The decomposition leads to a Mayer-Vietoris sequence in cohomology relating the cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) to the cohomology of the genus two mapping class group Γ 2 0 \Gamma _2^0 . Using this tool, the 3 3 - and 5 5 -primary pieces of the integral cohomology of S p ( 4 , Z ) Sp(4,\mathbb {Z}) are computed.

Smooth non-special surfaces of ℙ4

We consider smooth non-degenerate surfaces in ℙ4, and prove that there is a finite number of such surfaces which are: (a) sectionally non-special, i.e.h1(OC(1))=0, where C is a general hyperplane section of S; or (b) not of general type and non-special (i.e. h1(OC(1))=0.

A uniqueness theorem for automorphisms of a nondegenerate surface in a complex space

Consider a collection of k (where k>/1) real analytic real-valued functions ~i ..... , ~ defined in a neighborhood of some point 6 of a complex linear space. Suppose that ~i • (~) = ... = ~(~) = 0 and that the k vectors grad ~'(~) ..... grad ~k (~) are linearly independent as vectors of the complex linear space. Then the relations ~i .... = ~ = 0 define in a neighborhood of $ a real analytic surface M of codimension k and we have ~E/~f.

Electron energy level crossings in the time-dependent Born-Oppenheimer approximation

We summarize the results of a mathematical study of the time-dependent Born-Oppenheimer approximation near crossings of two non-degenerate electron energy surfaces. We illustrate our techniques by relatively simple examples that contain the essential ingredients of the general cases. We discuss all generic types of crossings of two non-degenerate electron energy surfaces.

Minimal immersion of surfaces in quaternionic projective spaces

For a minimal immersion of a surface in a quaternionic Kähler manifold a concept of non-degeneracy is defined. Then using a theorem on elliptic differential systems we show a non-degenerate surface is in a sense generic, and around each point with possible exception of an isolated set of degenerate points we can define a smooth Darboux frame. The frame is continuous at a degenerate point. Next, by reducing the structure group we define a symmetric sextic form of type $(6,0)$ and we show in the case that ambient space is $HP^{n}$ this form is a holomorphic (abelian) differential. The last section is a brief note on the relation of our work to Glazebrook's twistor spaces for $HP^{n}$.

The approximation of non-degenerate offset surfaces

This paper describes algorithms for approximating the offsets to general piecewise parametric surfaces by networks of bicubic patches. The surface to be offset may be composed of arbitrary parametric patches, provided precise tangent continuity obtains across all patch boundaries and the surface metric is everywhere non-singular (i.e., a unique surface normal is defined). The approximation scheme consists of three stages: (1) A differential surface analysis is performed to ascertain the extremum principal radii of curvature for each patch — this constrains the offset magnitude that may be specified without degeneration of the offset. (2) The parametric domain of each patch is sub-divided, and a bicubic approximant to the offset for each sub-domain is computed. (3) A tolerance analysis is performed on each patch of the offset approximation to evaluate its extremum deviations from the true offset. The accuracy of the offset approximation increases with the degree of subdivision of the parametric domain of each patch. Fractional errors in the range 10 −2 to 10 −3 are typical for two- or three- fold sub-division of each parametric variable (four- and nine- fold area division respectively), provided the surface is smooth and its smallest concave radius of curvature considerably exceeds the specified offset magnitude. A fully automatic tolerance-based surface offset capability may be developed by providing feedback between stages (3) and (2), successive degrees of parametric sub-division being determined by the errors from prior approximations until the desired accuracy is achieved.

Nondegenerate surfaces of degree n+3 in PnC.

Nondegenerate surfaces of degree n+3 in PnC.

Analytical approximate model of the potential distributions in semiconductor samples having a nondegenerate surface accumulation layer

A satisfactory analytical approximate model of the potential distribution (u) in a semiconductor sample having a nondegenerate surface accumulation layer possessing an arbitrary surface potential and a constant impurity concentration N≳5×101 ni has been obtained by means of joining the two different approximate expansions of the electrical field in terms of ue−u for large values of u and in terms of u for small values of u. The approximation deviates from the numerical solution by less than 1% through the whole sample.