Affine Kettengeometrien über Jordanalgebren

It is shown that an affine chain geometry over a Jordan algebra can be constructed in a nearly classical manner. Conversely, such chain geometries are characterized as systems of rational normal curves having a group of automorphisms with certain properties.

Herrn Prof. Dr. Heinz Lüneburg zu seinem 60. Geburtstag     

Chain Geometry Determined by the Affine Group

Chain geometry associated with an affine group and with a linear group is studied. In particular, closely related to the respective chain geometries affine partial linear spaces and generalizations of sliced spaces are defined. The automorphisms of thus obtained structures are determined.     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.     

Affine Barbilian-Ebenen II

Journal of Geometry - In part I we proved that every affine Barbilian plane is up to isomorphisms an affine geometry over a Z-ring R as defined in the introducing abstract there. Now we carry out...     

A generalized affine isoperimetric inequality

A purely analytic proof is given for an inequality that has as a direct consequence the two most important affine isoperimetric inequalities of plane convex geometry: The Blaschke-Santaló inequality and the affine isoperimetric inequality of affine differential geometry.     

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.     

Affine geometry of modules over a ring with invariant basis number

The fundamental theorem of affine geometry over rings with invariant basis numbers is considered.     

Extending the Concept of Chain Geometry

We introduce the chain geometry (K,R) over a ring R with a distinguished subfield K, thus extending the usual concept where R has to be an algebra over K. A chain is uniquely determined by three of its points, if, and only if, the multiplicative group of K is normal in the group of units of R. This condition is not equivalent to R being a K-algebra. The chains through a fixed point fall into compatibility classes which allow to describe the residue at a point in terms of a family of affine spaces with a common set of points.     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

Geometrical maps in ring affine geometries

The aim of this work is to investigate the validity of the fundamental theorem of affine geometry over rings with an invariant basis number.     

An intrinsic characterization of developable surfaces

We consider the classical theorem saying that if f: M → R³ is a Riemannian surface in R³ without planar points and with vanishing Gaussian curvature, then there is an open dense subset M′ of M such that around each point of M′ the surface f is a cylinder or a cone or a tangential developable. As we shall show below, the theorem, in fact, belongs to affine geometry. We give an affine proof of this theorem. The proof works in Riemannian geometry as well. We use the proof for solving the realization problem for a certain class of affine connections on 2-dimensional manifolds. In contrast with Riemannian geometry, in affine geometry, cylinders, cones as well as tangential developables can be characterized intrinsically, i.e. by means of properties of any nowhere flat induced connection. According to the characterization we distinguish three classes of affine connections on 2-dimensional manifolds, i.e. cylindric, conic and TD-connections.     

Some relations between riemannian and affine geometry

The aim of the paper is to characterize metric normals in terms of affine geometry and derive from that some consequences for affine geometry. Also, an affine affine version of the theorema egregium is proved.This research was supported by AvHumboldt Stiftung.     

Affine Lattices of Power Groups

For Hall power groups over a ring, the lattice of cosets is constructed. For class-2 nilpotent groups over the field, the fundamental theorem of affine geometry is proved. The given example shows that the theorem is invalid for the class of nilpotency ≥3.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Integral geometry in affine and projective spaces

A survey is given of results of integral geometry related to the integration of sections of one-dimensional bundles and differential forms over planes in affine and projective spaces.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 16, pp. 53–226, 1980.     

One dimensional metrical geometry

One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this situation, such as the Triple quad formula, the Triple spread formula and the Spread polynomials, which are universal analogs of the Chebyshev polynomials of the first kind. Chromogeometry appears here, and the related metrical and algebraic properties of the projective line are brought to the fore.      

Fundamental Theorem of Affine Geometry for Lie Algebras

For a Lie algebra over a ring, the lattice of cosets is constructed. Necessary and sufficient conditions for the distributivity, modularity, and semimodularity of coset lattices are found. The fundamental theorem of affine geometry for nilpotent Lie algebras of class 2 is proved.     

The Non-Collinearity Graph of the O+(8,2) Quadric Is Uniquely Geometrisable

The graph of the titlehas the points of the O⁺(8,2) polar space as itsvertices, two such vertices being adjacent iff the correspondingpoints are non-collinear in the polar space. We prove that, uptoisomorphism, there is a unique partial geometry pg(8,7,4)whose point graph is this graph. This is the partial geometryof Cohen, Haemers and Van Lint and De Clerck, Dye and Thas. Ouruniqueness proof shows that this geometry has a subgeometry isomorphicto the affine plane of order three, and the geometry is canonicallydescribeable in terms of this affine plane.     

Geometries involving affine connections and general linear connections

Summary Part I of this paper is concerned with the theory of differential invariants of a symmetric affine connection and a general linear connection of theK?nig type. Part II deals with a geometry in which the components of the affine connection areChristoffel symbols, and the general linear connection is of a special sort. This section can be considered in part as an extension of theEinstein-Mayer (5) geometry,