Rolf Walter

Constant mean curvature tori with spherical curvature lines in noneuclidean geometry

A previous result in Euclidean geometry [7] on H-tori with plane and spherical curvature lines is extended here to the two noneuclidean geometries. There result families of H-tori with only spherical curvature lines, which are explicitly representable by elliptic and theta functions (or ordinary integrals of elementary functions). Among the geometric properties, it is shown that the midpoints of the generating spheres vary on geodesics. The hyperbolic case is more similar to the Euclidean situation than the elliptic one. In elliptic geometry the constructed surfaces depend on one additional rational parameter and, as a limiting case, there are even countably many minimal tori of this type.

Nonnegativity of the curvature operator and isotropy for isometric immersions

Positivity of curvature plays a crucial role in riemannian geometry. However, such positivity conditions can be expressed in different ways of varying strength. In the more qualitative parts of riemannian geometry, the sectional curvature K and its sign is prevailing. The reason is that K enters the fundamental formulas for the geodesic variation in a very direct manner. For other questions, the curvature operator p and its definiteness is more dominant. For example, the positive semidefiniteness p > 0 is the only known universal condition ensuring the nonnegativity of the general Gauss-Bonnet integrand, while K > 0 is merely sufficient for dimensions less than 6. Moreover, for noncompact complete manifolds the Gauss-Bonnet integral and the Euler characteristic, if existing at all, do in general not coincide, but their difference is in some instances manageable if the curvature operator is positive semidefinite (see Theorem (2.F) below). It is well known that p >0 implies K > 0 , the converse being true only for dimensions less than 4. So, a natural question is What conditions have to be added to conclude from K > 0 that p > 0? Some known results in the realm of intrinsic geometry are listed in Sect. 2. Now, if a riemannian manifold M of dimension m is isometrically immersed in another riemannian manifold ~/ of dimension rh, we may conjecture that, for small codimensions p, = N m , there are weaker conditions that in the abstract case to ensure the positive semidefiniteness of p. The purpose of the present paper is to initiate some answers in this direction. For hypersurfaces (p =1) of euclidean space, p > 0 is directly implied by K > 0 since p equals the exterior square of the Weingarten map. For submanifolds of euclidean space with codimension p=2 , Weinstein [-31] deduced p > 0 from K > 0 . In Sect. 3 we generalize this result to the semidefinite case and to nonflat ambient spaces. Also, an application to the Gauss-Bonnet formula is given. However, there is no hope to go beyond p = 2 in this manner as is shown by a counterexample at the end of Sect. 3. For higher codimensions, an attack is then made in Sect. 4

Homogeneity for Surfaces in Four-Dimensional Vector Space Geometry

We classify all surfaces in R4 which are homogeneous in the sense of equi-centroaffine differential geometry. There result 21 group classes, some of them depending on one or two real parameters. The classification is cleared up, i.e. each copy is equivalent to exactly one representative. This applies as well to the corresponding groups as to the orbits (and also to the parameter cases). In particular, we can characterize the Clifford tori in a purely affine manner and determine all homogeneous centroaffine spheres. This answers a former question on the existence of centroaffine spheres which are not contained in a hyperplane. The classification and, in particular, the uniqueness is based on geometric insight and is essentially not computer dependent. The leading ideas are of a general nature and may also be applied to homogeneity for higher-dimensional cases and for related geometries.

Visibility of surfaces via differential geometry

Abstract Using the means of differential geometry, we develop a theory for rendering surfaces with complete visibility claryfying. The results are based on the new concept of ‘sight indices’ which control the change of coverings dynamically along arbitrary given curves. The method is entirely curve oriented, so suitable for most graphic outputs, in particular for plotters. Important attributes of surfaces, like silhouette, selfintersection, and boundary curves will be generated and, in fact, be viewed as constitutive for the whole process.

On the Role of the Burstin-Mayer Metric for Surfaces in R

For submanifolds of the affine space Rn, it is very important to derive a riemannian or pseudo-riemannian metric on the manifold just from affine data of the configuration. It is in this way that the equi-affine hypersurface theory is initiated by the so called Blaschke-Berwald metric (for the most recent state of affine hypersurface theory see the book of Li-Simon-Zhao [1993] and the vast literature given there). The same is true for the centro-affine geometry of codimension-two submanifolds (cf. Walter [1988], [1991 a]). Another instance where such a metric has been constructed from affine data are the (two-dimensional) surfaces of R4 (Burstin-Mayer [1927]). Recently, the geometry of these surfaces has been taken up by Nomizu-Vrancken [1993] with respect to the construction of a new transversal plane bundle. In the present note, we deal with the existence and, in particular, non-existence of elliptic points of the Burstin-Mayer metric from a local and global viewpoint.

Gauge Theory, Isotropy, and Surfaces in Affine 4-Space

The surface theory in the equiaffine space R4 is developed on the basis of H. Weyl’s gauge theory. Rescaling of the Weyl geometry leads to a 1-parameter family of invariant transversal plane bundles containig former special constructions. A transversal bundle metric is gained via the notion of isotropy. The paper then proceeds with a general tensorial theory, including theorema egregium and Radon-type results and a discussion of cubic fundamental forms. Finally there is given an application to homogeneous surfaces.

Anhang über Logik und Mengenlehre

Vom Leser werden nur minimale Vorkenntnisse inhaltlicher Natur erwartet. In diesem Anhang stellen wir einige Voraussetzungen formalen Charakters zusammen. Auf eine strenge Behandlung mus hier verzichtet werden. Der Leser kann jedoch die Grundaussagen, die ohne Beweis angegeben werden, als Axiome auffassen. Ausfuhrliche und gut verstandliche Darstellungen findet man zur Logik bei Quine und zur Mengenlehre bei Halmos [2]. Es soll allerdings nicht verschwiegen werden, das in diesem Zusammenhang Grundlagenfragen bestehen, die ihrer endgultigen Klarung noch harren.

Quadratische Hyperflächen in der affinen und euklidischen Geometrie

Wir studieren hier erstmalig nichtlineare Gebilde, die sog. quadratischen Hyperflachen oder Quadriken. Die Untersuchung erfolgt zunachst im Rahmen der Affingeometrie eines K-Vektorraumes V, spater wird die euklidische Situation beleuchtet.

Reelle Räume mit Skalarprodukt

Die Vektorraume, die in den Anwendungen und in anderen Gebieten der Mathematik auftreten, besitzen meistens eine Zusatzstruktur metrischer oder topologischer Natur, so das man Langen oder Umgebungen von Vektoren zur Verfugung hat (was in einem „nackten“ Vektorraum nicht der Fall ist). Wir behandeln hier die Zusatzstruktur „Skalarprodukt“. Euklidische Vektorraume, die in 0.3 motiviert wurden, sind z.B. reelle Vektorraume mit einem positiv definiten Skalarprodukt.

Aus der Algebra

Die Algebra, die fruher einmal die Kunst der Gleichungsauflosung war, ist heute in eine allgemeine Theorie der Verknupfungen eingemundet, die zumeist abstrakt, also axiomatisch betrieben wird. Die axiomatische Denkweise beruht auf der folgenden Grundidee: Man stellt zunachst fest, das gewisse Gesetzmasigkeiten fur unterschiedliche Objekte gleichermasen gelten. Daraufhin lost man sich von den konkreten Gultigkeitsbereichen, indem man solche Gesetze zu definierenden Eigenschaften einer neuen Struktur erhebt. Dieses Vorgehen hat sich in der modernen Mathematik auserordentlich bewahrt. Einer seiner Vorzuge ist die geistige Okonomie, die es bewirkt: Erkenntnisse, die im abstrakten Rahmen gewonnen wurden, besitzen eine universelle Gultigkeit; sie brauchen in konkreten Fallen nicht erneut uberpruft zu werden. Unter den vielen algebraischen Strukturen, die so aufgebaut wurden, konzentrieren wir uns in diesem Kapitel auf die Gruppen und Korper.