Poincare algorithm for cycle of k-dimensional face
by
Vitalii S. Makarov
Russian Peoples Friendship University, Moscow

Let polyhedron Mⁿ be in space Xⁿ of constant curvature (i.e. Xⁿ is the sphere Sⁿ, or the Euclidean space Eⁿ, or Lobachevsky space Hⁿ). Let facets of Mⁿ be pairwise identified by motions of the space Xⁿ which generate a crystallographic group \Gamma without torsions. The set of all k-dimensional faces of polyhedron Mⁿ which are equivalent to F^k subset Mⁿ by identification given is called the cicle of face F^k. (By identification the set of one cycle yields one face of manifold Mⁿ=Xⁿ/\Gammaⁿ.)

In our communication we state an algorithm that permits to describe the structure of nieghbourhood of k-dimensional face of manifold Mⁿ=Xⁿ/\Gamma (obtained from Mⁿ by identification of its facets). The way to do it is to describe the scheme of adjacencies of cells of spherical complex that arises by intersection of the sphere of small enough radius with centre in an inner point of face F^k and of the incident to the centre of the sphere orthogonal complement to the face F^k. The algorithm consists in subsequent construction of crowns of any cell of spherical complex with using induction in co-dimension of face.

Some examples of using algorithm when studying geometry of manifolds of constant negative curvature are given (investigation of isometry groups of manifolds, geodeticly embedded submanifolds, regular star polytopes, hiperbolic manifolds and polyhedron etc.)

Date received: March 10, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caet-28.