Möbius Geometry with Mathematica

In a series of papers my coworkers Ch. Schiemangk, Ch. Dittrich and I considered the n-dimensional Möbius Geometry: the conformal geometry of the n-dimensional sphere with its Riemannian metric of constant curvature. A survey of our work (in German) is given in the subsection Möbius Geometrie of this homepage. In Part 1 of the present article I describe the Mathematica programs I developed to calculate the Möbius invariants of the basic configurations of k-dimensional subspheres of the n-dimensional sphere. Part 2 contains a report about my Mathematica notebooks and packages devoted to the differential geometry of curves in the 3-dimensional Möbius space or the Möbius plane. I hope to be able to continue this work applying Mathematica to the differential geometry of surfaces and higher dimensional submanifolds of the n-dimensional Möbius space.

1. Elementary Möbius Geometry

In this part I consider the simplest elementary objects of Möbius geometry. the points and the k-dimensional subspheres of the n-dimensional Möbius space, k = 0,1,...,n-1. The Möbius group acts transitively on the manifold of all subspheres of the same dimension k, but for pairs of subspheres exist Möbius invariants. The algebraic background can be found in my paper [33], see also Section 2.7 of the book [34], "Projective and Cayley-Klein Geometries". There one finds a complete systems of Möbius invariants for pairs of a k- and an l-sphere for arbitrary k, l = 0, 1,..., n-1. These invariants are similar to the stationary angles of a k- and an l-plane in Euclidean geometry. It is remarkable that for solving the problem it suffices to apply the method of the Principal Axis Theorem (Hauptachsentransformation) known in the theory of quadratic forms and quadrics in Euclidean geometry; remember that the n-dimensional Möbius space is algebraically modelled by the generating lines of the isotropic hypercone of a pseudo-Euclidean vector space of dimension n+2 and index 1.

The Möbius invariants of pairs of subspheres of the 3-dimensional Möbius space have been calculated and their properties are investigated with Mathematica in notebooks and packages collected under the title Spheres being obsolete now. The contents of "Spheres", in particular the three notebooks about pairs of subspheres, are contained now in a revised form, adapted to Wolfram Mathematica v. 11.3, in the Mathematica software collection "Elementary Möbius Geometry", elmoeb.zip, download here. To see the contents of the notebooks contained in elmoeb.zip click on the following links:

emg0.nb, Elementary Möbius Geometry. Initialization.

emg1.nb, Elementary Möbius Geometry I. Points and Spheres.

This notebook contains the basic concepts of real Möbius geometry. The Möbius group acts transitively on the n-dimensional sphere as a point manifold and on the manifolds of its k-dimensional subspheres. In the notebook the 3-dimensional geometry is treated; many tools are applicable also for calculations in higher dimensions. It contains the Möbius geometry of points and spheres.

emg2.nb, Elementary Möbius Geometry II. Circles.

This notebook contains the geometry of circles. First we introduce into Mathematica the circles as objects in the 3-dimensional Euclidean space; this is necessary since in Mathematica the built-in Circles are considered in the plane only. The manifold of all circles of the 3-dimensional space is a 6-dimensional symmetric pseudo-Riemannian space, on which the Möbius group acts transitively. We describe the relations of the Euclidean invariants and the Möbius invariants of the circles and of the circle pairs and give them a geometrical interpretation. In analogy to the Coxeter invariant the Möbius invariants of a circle pair are expressed by its Euclidean invariants.

emg3.nb, Elementary Möbius Geometry III. Pairs of Subspheres in S^3.

This notebook continues the notebooks Elementary Möbius Geometry I, II. We consider here those pairs of subspheres of the Möbius space not being treated in these notebooks.

2. Möbius Differential Geometry

Curve Theory

Curves in the n-dimensional Möbius space are treated in part I of a manuscript of a not yet ready book which can be downloaded here. Chapter 1 of the manuscript contains a description of the n-dimensional Möbius space as a homogeneous space of the pseudo-orthogonal Lie group O(1,n+1). Conformal models of the simply connected space forms of the Euclidean, spherical, and hyperbolic spaces of constant curvature are constructed. Using the structure forms of linear Lie groups the group spaces are considered as frame manifolds, and the linear homogeneous spaces as frame bundles. The basic geometric invariants are deduced applying the linear isotropy representations.

Chapter 2 gives a survey of E. Cartan's method of moving frames. With this method the Möbius structure of immersions of m-dimensional manifolds into the n-dimensional Möbius space is constructed. It contains a first raw classification of the immersions, tangential and normal bundle, its Möbius-invariant Riemannian metric, including volume measure and linear connections. This chapter is an elaboration of the first part of the paper [22].

Chapter 3 contains the theory of curves in the n-dimensional Möbius space. A generally curved curve posses a Frenet frame and a natural conformal parameter. It is characterized up to a Möbius transformation by n-1 curvatures. This has been proved by L. L. Verbitzkij in 1959 by tensor methods and in my paper [24] by E. Cartan's method. As an application we classify and describe all the curves of constant conformal curvatures in dimensions n = 2, 3. Finally we sketch a general curve theory in linear homogeneous spaces and characterize the curves of constant curvatures of these spaces as orbits of one-parametric subgroups.

The Mathematica notebooks on this subject accompany and confirm the presentation of the theory developed in the manuscript. Moreover they provide tools for the explicit calculation of the Möbius invariants of the curves. The generally curved plane curves of constant curvatures are the loxodromes. Together with a first attempt of programming Möbius curve theory in Mathematica they are treated in the notebook loxodromes. They are also considered in the notebook Curves in the Möbius Plane. The 3-dimensional curves of constant curvatures earlier contained in a now obsolete notebook Curves of Constant Curvatures in Möbius Geometry are now included as Section 4 in the notebook Curves in the Möbius Space. It is shown that all the generally curved 3D-curves of constant Möbius curvatures are also curves of constant Riemannian curvatures in one of the conformal models of the Riemannian spaces of constant curvature.

The interactive Mathematica notebooks can be downloaded here:


Curves in the Möbius Plane

Curves in the Möbius Space

For working with these notebooks some packages are necessary. They can be downloaded here: moebpack.zip.

These notebooks contain Modules for the calculation of Möbius-invariant Frenet frames, the natural parameter, and the conformal curvatures of the curves. Tools for calculating and plotting their osculating circles or spheres are developed and tested. The Frenet frames are moving isotropic-orthonormal frames in the corresponding 4- or 5-dimensional pseudo-Euclidean vector space of index 1, containing besides the isotropic vector function representing the points of the curve, a uniquely and equivariantly defined second isotropic vector function which defines a corresponding curve in the Möbius plane resp. space named the anti-curve of the given curve. Modules for calculating and plotting the anticurves are given and it is proved that the relation curve → anticurve is not involutive.

Revised: October 9, 2019.