Classification of surfaces with planar normal sections

A surface in E^m is said to have planar normal sections if normal sections of M are planar curves. In this paper we completely classify surfaces in E^m with planar normal sections. Consequently, a new characterization of the Veronese surface is obtained.     

Isotropic submanifolds with pointwise planar normal sections

Submanifolds of E^m with pointwise planar normal sections were studied in [1] and others. In the present paper, we will prove that an isotropic submanifold in E^m with pointwise planar normal sections is isometric to a symmetric space of rank one or to a Euclidean space. Moreover we will determine such surfaces in E^m with the above assumptions.     

Differential geometry of submanifolds with planar normal sections

Summary Submanifolds of E^m with (pointwise) planar normal sections were studied in [3, 4]. In the present paper, we will prove that the first normal spaces of such submanifolds are always parallel in the normal bundle. Moreover, we will prove that a spherical submanifold has parallel second fundamental form if and only if it has pointwise planar normal sections. Some applications of these fundamental results will also be given.     

Surfaces in a pseudo-euclidean space with planar normal sections

A pseudo-Riemannian surface M in a pseudo-Euclidean space E _s ^m is said to have planar normal sections if normal sections of M are planar curves. In the present paper we give some classification theorem concerning surfaces in E _m ^s with planar normal sections.     

The maximum size of the planar sections of random spheres and its application to metallurgy

A theorem of this paper proves that if the size distribution of random spheres is generalized gamma, its Wicksell transform and other related distributions belong to the domain of attraction of the Gumbel distribution. The theorem also shows the attraction coefficients of the distributions. The fatigue strength of high-strength steel is closely related to the maximum size of nonmetallic inclusions in the region of maximum stress of the steel. Murakami and others developed a method, making use of the Gumbel QQ-plot, for predicting the maximum size from the size distribution of inclusion circles in microscopic view-fields. Based on the Gumbel approximation of the maximum of wicksell transforms, a modified and extended version of Murakami's method is justified, and its performance is evaluated by simulation.     

Immersions with circular normal sections and normal sections of product immersions

We study immersions with normal sections that are circles in the ambient Euclidean space and formulate lemmas concerning normal sections of product immersions. As applications we determine all parallel immersions with planar normal sections and all immersions with planar normal sections and trivial normal connection.Aspirant N.F.W.O.     

Zhang's conjecture and squares of Abelian surfaces

We give in this Note some squares of Abelian surfaces that are counterexamples to a conjecture formulated by Zhang about the intersection of subvarieties and preperiodic points.     

Projective surfaces with bi-elliptic hyperplane sections

We study projective surfaces X which have a bi-elliptic curve (i.e. 2∶1 covering of an elliptic curve) among their hyperplane sections . We give a complete characterization of those surfaces when their degree d is d≥17 (only conic bundles and scrolls if d≥19, three possible exception otherwise) and when d≤8. A conjecture is given for the remaining cases. The main tool we use is the study of the adjunction mapping on X.     

Classification of marginally trapped Lorentzian flat surfaces in and its application to biharmonic surfaces

A surface in a semi-Riemannian manifold is called marginally trapped if its mean curvature vector field is light-like at each point. In this article, we classify marginally trapped Lorentzian flat surfaces in the pseudo-Euclidean space . As an application, we obtain the complete classification of biharmonic Lorentzian surfaces in with light-like mean curvature vector.     

Absolute planar retracts and the four color conjecture

The Four Color Conjecture (4CC) is shown to be equivalent with the existence of absolute retracts in the class $\text{P}$ of all planar graphs and also with some other conditions on the class $\text{P}$. Assuming the 4CC is true we determine all absolute planar retracts. A possible argument for the decidability¹ of the 4CC is discussed and proved to be equivalent to the 4CC itself.     

Classification of five-dimensional generalized pointwise symmetric Riemannian spaces

Riemannian symmetric spaces have the following two classes of spaces as their natural generalizations: (A) the class ofGS-spaces (generalized symmetric Riemannian spaces); (B) the class ofGPS-spaces (generalized pointwise symmetric Riemannian spaces). A result due to O. Kowalski says that the relation between the two classes is (A) (B), the inclusion being strict. In the present paper the author proves that in dimension 5 the class (A) and the class (B) must coincide. Consequently the explicit classification of five-dimensional GPS-spaces is obtained.     

Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures

Wintgen proved (C. R. Acad. Sci. Paris, 288:993–995, 1979) that the Gauss curvature K and the normal curvature K ^D of a surface in Euclidean 4-space \mathbb E⁴{\mathbb {E}^4} satisfy K + |K ^D | ≤ H ², where H ² is the squared mean curvature. A surface in \mathbb E⁴{\mathbb {E}^4} is called Wintgen ideal if it satisfies the equality case of the inequality identically. Wintgen ideal surfaces in \mathbb E⁴{\mathbb {E}^4} form an important family of surfaces, namely, surfaces with circular ellipse of curvature. In this article, we completely classify Wintgen ideal surfaces in \mathbb E⁴{\mathbb E^4} satisfying |K| = |K ^D | identically.     

On singly-periodic minimal surfaces with planar ends

The spaces of nondegenerate properly embedded minimal surfaces in quotients of 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.

     

Infinitesimal nonrigidity of convex surfaces with planar boundary

In the present paper infinitesimal nonrigidity of a class of convex surfaces with planar boundary is given. This result shows that if the image of the Gauss map of an evolution convex surface with planar boundary covers some hemisphere, this surface may be of infinitesimal nonrigidity for the isometric deformation of planar boundary.     

Filling area conjecture and ovalless real hyperelliptic surfaces

We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We translate the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredients. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real surfaces of even positive genus. Here the singular points are Weierstrass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics.Received: February 2004 Revised: February 2005 Accepted: November 2004     

Torsion sections of elliptic surfaces

Given a torsion section of a semistable elliptic surface we prove equidistribution results for the components of singular fibers which are hit by the section and for the root of unity (identifying the zero component withC) which is hit by the section in case the section hits the zero component Research supported in part by the NSF under grant DMS 9104058 and the NSA under grant MDA 904-92 H 3022     

On complex projective surfaces with trigonal hyperplane sections

Let S be a complex projective surface endowed with an ample and spanned line bundle L. Assume that (S,L) does not belong to some special classes and that c_l(L)²10. We prove that(K_SL)·K_S–3 and |L| contains a trigonal curve (of genus4) iff either (S,L) is a rational surface ruled by cubics, or the g¹ ₃ of C is cut out by |K_S ^–1|. This result applies to surface having a hyperplane section which is a trigonal curve.Partially supported by the M.P.I. of the Italian Government     

Thin position and essential planar surfaces

Abby Thompson proved that if a link is in thin position but not in bridge position, then the knot complement contains an essential meridional planar surface, and she asked whether some thin level surface must be essential. This note is to give a positive answer to this question, showing that if a link is in thin position but not bridge position, then a thinnest level surface is essential. A theorem of Rieck and Sedgwick follows as a consequence, which says that thin position of a connected sum of small knots comes in the obvious way.