This chapter discusses isoparametric hypersurfaces and their generalizations The chapter presents a survey of isoparametric hypersurfaces, discusses Dupin hypersurfaces, and describes isoparametric submanifolds in ambient spaces that are finite dimensional Euclidean spaces or infinite dimensional Hilbert spaces The taut submanifolds in Riemannian manifolds are described The classification of isoparametric hypersurfaces in spheres with three principal curvatures is related to the various characterizations of the standard embeddings of the projective planes The chapter discusses isoparametric hypersurfaces in spheres with four different principal curvatures, all of which are assumed to have the same multiplicity The Clifford examples of Ferus, Karcher, and Munzner together with the homogeneous hypersurfaces are all known examples of isoparametric hypersurfaces in spheres A difference between isoparametric and Dupin hypersurfaces is that although the parallel hypersurfaces of the Dupin ones are also Dupin, they do not foliate the ambient space as the isoparametric ones do The hypersurface of a sphere is isoparametric if and only if it is equifocal

Chapter 10 - A Survey on Isoparametric Hypersurfaces and Their Generalizations

The monograph deals with the theory of conformal, geodesic,
holomorphically projective, F-planar and others mappings and
transformations of manifolds with affine connection,
Riemannian, Kahler and Riemann-Finsler manifolds. Concretely,
the monograph treats the following: basic concepts of
topological spaces, the theory of manifolds with affine
connection (particularly, the problem of semigeodesic
coordinates), Riemannian and Kahler manifolds (reconstruction
of a metric, equidistant spaces, variational problems in
Riemannian spaces, SO(3)-structure as a model of statistical
manifolds, decomposition of tensors), the theory of
differentiable mappings and transformations of manifolds (the
problem of metrization of affine connection, harmonic
diffeomorphisms), conformal mappings and transformations
(especially conformal mappings onto Einstein spaces, conformal
transformations of Riemannian manifolds), geodesic mappings
(GM; especially geodesic equivalence of a manifold with affine
connection to an equiaffine manifold), GM onto Riemannian
manifolds, GM between Riemannian manifolds (GM of equidistant
spaces, GM of Vn(B) spaces, its field of symmetric linear
endomorphisms), GM of special spaces, particularly Einstein,
Kahler, pseudosymmetric manifolds and their generalizations,
global geodesic mappings and deformations, GM between
Riemannian manifolds of different dimensions, global GM,
geodesic deformations of hypersurfaces in Riemannian spaces,
some applications of GM to general relativity, namely three
invariant classes of the Einstein equations and geodesic
mappings, F-planar mappings of spaces with affine connection,
holomorphically projective mappings (HPM) of Kahler manifolds
(fundamental equations of HPM, HPM of special Kahler manifolds,
HPM of parabolic Kahler manifolds, almost geodesic mappings,
which generalize geodesic mappings, Riemann-Finsler spaces and
their geodesic mappings, geodesic mappings of Berwald spaces
onto Riemannian spaces.

Differential Geometry of Special Mappings

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. 
As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.

/pdf/multitriangulations-pseudotriangulations-and-primitive-1v8my5qssn.pdf

Multitriangulations, pseudotriangulations and primitive sorting networks

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial.

As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.

/pdf/multitriangulations-pseudotriangulations-and-primitive-3r49b4x4fl.pdf

Multitriangulations, Pseudotriangulations and Primitive Sorting Networks

The axiomatics of ordered geometry: I. Ordered incidence spaces

A topological translation plane may be defined as a system L of a‰ne l-dimensional subspaces of R 2l such that (1) any two points are joined by a unique line L A L and joining is continuous; (2) Euclid's parallel postulate holds and L is invariant under translation. It is easy to see that one of the two parts of condition (2) may be omitted. We use topological methods to show that (2) may be omitted altogether, unless l ¼ 8. In other words, we show that, except possibly for l ¼ 8, there are no proper stable planes with point set R 2l and a‰ne subspaces as lines, and that translation invariance is automatic. The core of the argument is Theorem 3.1, which characterizes the lines of a topological translation plane among a‰ne subspaces: an l-dimensional subspace A is a line if there is a point p B A such that every line passing through p intersects A in at most one point. We also consider the eect of the dual spread condition in our context. Moreover, we point out limits to generalization by giving examples obtained by transfinite induction in finite dimensional vector spaces over arbitrary infinite skew fields.

/pdf/affine-line-systems-in-finite-dimensional-vector-spaces-pxlzpv782y.pdf

Affine line systems in finite-dimensional vector spaces

By constructing all 4-dimensional affine Hughes planes that are not translation planes, we contribute the last missing part to the classification of all compact 4-dimensional projective planes having a non-solvable automorphism group of dimension at least 6. We use the information on the possible actions of the automorphism group obtained by the third author [5] and the construction principle employing derived planes that was given by the second author [4]. The unique solvabilty of certain nonlinear systems of equations has to be shown; this is done using topological degree techniques and delicate analytic arguments. Also the proof of continuity of the geometric operations is quite subtle.

Four-dimensional compact projective planes admitting an affine Hughes group

On the definition of topological polygons

We introduce topological parallelisms of oriented lines (briefly called oriented parallelisms). Every topological parallelism (of lines) on $${\mathrm{PG}(3,{\mathbb {R}})}$$
 gives rise to a parallelism of oriented lines, but we show that even the most homogeneous parallelisms of oriented lines other than the Clifford parallelism do not necessarily arise in this way. In fact we determine all parallelisms of both types that admit a reducible $${\mathrm{SO}_3 {\mathbb {R}}}$$
 -action (only the Clifford parallelism admits a larger group (Lowen in Innov Incid Geom. arXiv:1702.03328
 ), and it turns out surprisingly that there are far more oriented parallelisms of this kind than ordinary parallelisms. More specifically, Betten and Riesinger (Aequ Math 81:227–250, 2011) construct ordinary parallelisms by applying $${\mathrm{SO}_3 {\mathbb {R}}}$$
 to rotational Betten spreads. We show that these are the only ordinary parallelisms compatible with this group action, but also the ‘acentric’ rotational spreads considered by them yield oriented parallelisms. The automorphism group of the resulting (oriented or non-oriented) parallelisms is always $${\mathrm{SO}_3 {\mathbb {R}}}$$
 , no matter how large the automorphism group of the non-regular spread is. The isomorphism type of the parallelism depends not only on the isomorphism type of the spread used, but also on the rotation group applied to it. We also study the rotational Betten spreads used in this construction and their automorphisms.

Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

We conjecture that the automorphism group of a topological parallelism on real projective 3-space is compact. We prove that at least the identity component of this group is, indeed, compact.

Rainer Löwen

Papers

Affine line systems in finite-dimensional vector spaces

Four-dimensional compact projective planes admitting an affine Hughes group

On the definition of topological polygons

Rotational spreads and rotational parallelisms and oriented parallelisms of $${\mathrm{PG}(3,{\mathbb {R}})}$$ PG ( 3 , R )

Compactness of the automorphism group of a topological parallelism on real projective 3-space