Numerical approximation of curve evolutions in Riemannian manifolds

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean plane. Examples include the hyperbolic plane, the hyperbolic disc and the elliptic plane, as well as any conformal parameterization of a two-dimensional manifold in ${{\mathbb{R}}}^d$, $d\geqslant 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate schemes for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${{\mathbb{R}}}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations.

Bireflectionality in Classical Groups

The motion groups of the real Euclidean plane and of the elliptic plane, the group of projectivities of a line, the projective general linear group PGL2(K), some orthogonal groups O3(K, Q) with char K = 2 (see [8]), are all bireflectional (zweispiegelig). There can be no doubt that bireflectional groups are of prime importance in any theory of groups that are generated by involutions. A brief look into F. Bachmann's book [1] gives convincing evidence.