This Section contains some contributions or comments to themes of differential geometry.
The Fundamental Theorem for Curves in the n-Dimensional Euclidean Space.
The existence- and uniqueness theorem for a curve whose curvatures are
given as functions of its arclength is the key tool for the curve theory in
elementary Euclidean differential geometry. Unfortunately, I don't know a
textbook that contains the necessary conceptual framework and a precise
complete proof. For understanding the programs developed under St. Wolfram's
Mathematica in my notebook "Euclidean Curve Theory" and the corresponding Mathematica
packages such a theoretical background is useful, if not necessary. This is
the reason for writing this paper, which may be read independently of the
mentioned Mathematica tools.
The notebook contains programs for the calculation
of the Euclidean invariants of curves in arbitrary dimension and for
graphical presentations in dimensions 2 and 3. Together with the needed
packages it can be downloaded here .
Differential Geometry of the Möbius Space I,
Curve Theory
Here part I of a manuscript of a not yet ready book can be downloaded.
Chapter 1 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 volumen 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 posseses a Frenet frame and a natural conformal parameter and is characterized up to a Möbius transformation by n-1 curvatures. This has been proved by L. L. Verbitzkij in 1959 by tensorial 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 scetch 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 calculations and many details of the curves of constant curvatures are given in two Mathematica notebooks. The curve theory of the Möbius plane is considered in the notebook loxodromes and, more complete, in the notebook Curves in the Möbius Plane; the 3-dimensional case is treated in the notebook Curves of Constant Curvatures in Möbius Geometry . The notebooks are written in St. Wolfram's computer-algebraic program Mathematica. The interactive Mathematica notebooks can be downloaded here:
Curves of Constant Curvatures in Möbius Geometry
For working with these notebooks some packages are necessary. They can be downloaded here:
Möbius Geometry, a Report
This is a survey about my papers to this field (in German).
Letzte Aktualisierung: 27.03.2012