Geometry and Algebra

Projective and Cayley-Klein Geometries

Under the influence of my teachers Hans Reichardt and W. Blaschke my mathematical studies have been guided by the ideas of Sophus Lie and Felix Klein, in particular by the Erlanger Program. In the studying year 1961/62 I visited the Moscow State University and listened lectures of P. K. Raschewski, A. L. Onishchik, E. B. Dynkin and others. In this time arose my friendship. with Arkady Onishchik leading to a fruitful cooperation over many years, and finally to our monography

Projective and Cayley-Klein Geometries

To get more information about this book click the title or here.

The German source of this translation can be seen and downloaded:

Deutsche Version (Berlin 2005)

Projektive und Cayley-Kleinsche Geometrien

 File geo.pdf (5,7 Mb)

Dieses File kann unmittelbar eingesehen und abgespeichert werden. Das Vorwort und eine Bemerkung zur Entstehung des Werkes findet man hier.

Für Hinweise auf Fehler, auch auf Druckfehler, sind die Autoren dankbar. Bitte an

sulanke@mathematik.hu-berlin.de senden.

Elementary Euclidean Geometry

Wolfram's Mathematica contains all linear algebraic tools necessary for working in Euclidean Geometry. The notebook “Euclidean Geometry” shows these tools and some variations of them and applies them to define some functions useful in particular in conformal Euclidean geometry. Here you can see a survey of the contents: Euklidsh.pdf . Obviously, the notebook may not and will not replace a textbook about Euclidean geometry. I hope that it will be useful for students of the first course who want learning linear algebra and geometry and be introduced into Mathematica. You may download the notebook Euklid.nb together with the package vectorcalc.m, containing affine geometric tools, the package euvec.m, containing modules belonging to Euclidean geometry, and a protocol file rank1.txt, here. Finished January 30, 2010

Elementary Möbius Geometry

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. Here I mention only the algebraic part of our publications. They consist in Section 2.7 of our book Projective and Cayley-Klein Geometries, the paper [33], and some Mathematica notebooks and packages collected under the title Spheres being obsolete now. The main content of this collection is contained now in three Mathematica notebooks titled Elementary Möbius Geometry I - III, and some Mathematica packages available below .

The fundamental objects of n-dimenional Möbius geometry are the points, circles, spheres, more general the k-dimensional subspheres (shortly k-spheres). In the paper

[33] Möbius invariants for pairs (S_1^m, S_2^l) of spheres in the Möbius space S^n. 
Beiträge zur Algebra und Geometrie, 41 (2000), No. 1, 233-246. SFB 288, Preprint Nr. 373, Berlin 1999., math.MG/9902131 

the problem to find complete systems of Möbius invariants for pairs of a k- and an l-sphere for arbitrary k, l = 0, 1,..., n-1 is solved. 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 component analysis (PCA)  (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.

Elementary Möbius Geometry I. Points and Spheres. (Mathematica Notebook)

This notebook named emg.nb contains the fundaments of real Möbius geometry. The Möbius group acts transitively on the n-dimensional sphere as a point manifold and on the manifold 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. A detailed survey of the notebook can be seen clicking here. The Mathematica code of the notebook can be downloaded: emg.nb.

Elementary Möbius Geometry II. Circles. (Mathematica Notebook)

contains the geometry of circles. First we introduce in Mathematica the circles as objects in the 3-dimensional Euclidean space; this is necessary since the built-in Circles are considered in the plane only. The manifold of all circles of the 3-dimensional space is a 6-dimenional 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.A detailed survey of the notebook can be seen clicking here. The Mathematica code of the notebook can be downloaded: emg2.nb.

Elementary Möbius Geometry III.

Pairs of Subspheres in S^3 .

(Mathematica Notebook)

This notebook continues the notebooks Elementary Möbius Geometry I, II. We consider here those pairs of subspheres of the Möbius space being not treated in these notebooks. A detailed survey of the notebook can be seen clicking here. The Mathematica code of the notebook can be downloaded: emg3.nb.

To work interactively with the notebooks and for your own applications the Mathematica packages contained in moebpack.tar.gz are needed, download. For older versions and the relations to other notebooks see here.

The contents of moebpack.tar.gz:

In a talk at the Jahrestagung der DMV 2011 in Köln I presented the following Examples of applictions of these packages:

1. Pseudo-Euclidean Orthogonalization.

2. Spheres and the Coxeter Invariant.

3. Sphere through Four Random Points.

4. Circle through Three Random Points.

5. Tubes of Torus Knots.

6. Euclidean and Möbiius Invariants of Circle Pairs.

7. Curves of Constant Möbius Curvatures.

8. Dupin Cyclides.

The talk can be deen here. The Mathematica notebook containing the examples can be downloaded here. Of course, working interactively with the notebook needs the packages moebpack.tar.gz, download.

Symplectic Geometry

It is well known that symplectic scalar products, i. e. non degenerated skew symmetric bilinear forms, exist only in even-dimensional vector spaces. A vector space is called symplectic, if a symplectic scalar product for its vectors is distinguished. The symplectic group is defined as the group of linear transformations of the vector space preserving the symplectic scalar product; it acts transitively on the corresponding odd-dimensional projective space, defining the projective symplectic geometry as the theory of geometric properties invariant under this action, in the sense of F. Klein's Erlanger Programm. Since in dimension 2 the symplectic group coincides with the special linear group, the symplectic geometry of a projective line coincides with its projective geometry. The first interesting case is the three-dimensional one, the main subject of the notebook symplectic.nb. Closely related to the projective geometry is the spherical geometry. One easily verifies, that the symplectic groups act transitively also on the odd-dimensional spheres, being double coverings of the projective spaces of the same dimension. The tools developed in this notebook can also be applied to explore the spherical symplectic geometries.

In the first section basic concepts of symplectic linear algebra are presented. An algorithm called symplectic orthogonalization seems to be new: it constructs for a given sequence of vectors a symplectic or optional an orthosymplectic sequence of vectors with the same span. This also gives a method to define adapted bases for subspaces. Rank and index of the scalar product restricted to the subspace are calculated, and the symplectic vector sequence consists of a basis of the defect subspace and a symplectic basis for a complementary symplectic subspace within the span.

The second section contains some considerations of symplectic transformations. In particular, we introduce a very simple class of these transformations, namely the symplectic transvections, which generate the symplectic group.

Section 3 is devoted to symplectic line geometry. The absolute of projective symplectic geometry is the complex of the isotropic lines, called the absolute null system. The complement of the nullsystem is the set of symplectic lines, on which the restriction of the scalar product does not vanish. (Remember that the projective lines are the two-dimensional vector subspaces.) For pairs of symplectic lines there exists a symplectic invariant being similar to the distance of two lines in metric geometries. The value of this invariant is the function sym defined and studied in this section.

The aim of the last two sections is the classification of the quadrics in the 3-dimensional complex or real projective symplectic spaces. As in Euclidean or affine geometries one classifies the symmetric bilinear forms whose corresponding quadratic forms define the quadrics. They are equivariantly associated with a special class of endomorphisms, the skew symmetric operators, of the underlying vector space. These operators can be classified using their Jordan decompositions. In section 4 this is done for complex symplectic spaces; for each class normal forms of the operators and the bilinear forms are found. Section 5 describes the refinements necessary for real spaces. Also in this case normal forms are obtained with the help of which one may discuss the shape of the quadrics.

The appendix contains modules useful for any application of Mathematica to linear algebra and the corresponding geometries. They are contained in the package vectorcalc.m. The modules specific for symplectic linear algebra are collected in the package symplecticgeo.m. They may be downloaded from this homepage.

Lie Algebras

For the classical series of Lie algebras over the real or the complex numbers sometimes extensive calculations of basic invariants like Killing form or index are necessary. Some tools and results in this field are contained in the notebook liealgeb.nb, see further information

Last revision February 27, 2012