Homeomorphic transformations

The family of all symmorphy transformations of the given object p(r) form a subgroup gp of the group G of all homeomorphic transformations of the 3D space. This subgroup gp is, in fact, defined by the shape properties of the 3D object, and it provides a complete characterization of its shape, in our case, the shape of the molecular charge density function p(r). 

The family of all possible homeomorphisms of three-dimensional space is a group Evidently, any two such homeomorphisms applied consecutively correspond to one such homeomorphism (closure property). The unit element is the identity transformation. Each homeomorphism has an inverse, and the product of homeomorphisms is associative. 

The shape of a continuum A can be characterized by a subfamily Gsph(/1) of Gpo , where the subfamily Gsp, (/4) contain all those homeomorphisms S from G on, that bring the transformed object Sv4 into an arrangement that is indistinguishable from A. Since for such homeomorphisms S the morphologies of the transformed object and the original object A are indistinguishable, S is called a symmorphy transformation of object A. 

In the special case of selecting the three-dimensional object as a molecular charge density function p(r), the analogies between point symmetry and symmorphy are rather clear. By analogy with the point symmetry of nuclear arrangements, the molecular charge density function pit) can provide a criterion for selecting the symmorphy transformations of p(r) from the infinite family G, of homeomorphisms of the three-dimensional space. A homeomorphism 5 is a symmorphy transformation... 

Among the transformations in family G one finds all the symmetry operations, but also all reflections in curved mirrors, nonlinear stretchings, and all continuous distortions of the space. Evidently, all possible homeomorphisms of the 3D space form a group G. Any two such transformations applied consecutively correspond to one such transformation (closure property) the unit element is the... 

Although Hopf (1940) showed that it is not possible to represent the rotation group in a one-to-one global manner with less than five parameters, quaternions are usually used to parametrize the rotation in a two-to-one way. In general no difficulties arise with such a parametrization, because the correspondence between the quaternions of unit length and the elements of the rotation matrix is a local homeomorphism. Using the smallest number for a one-to-one correspondence, i.e. five parameters, does not lead not to simple equations and thus there is no apparent advantage in the reduction of the parameters by this method. Another convenient method is to use six parameters, e.g. the elements of the first two columns of the transformation matrix. 

The simplest and most important polyhedra are the so-called simple polyhedra, which can be continuously deformed into spheres (i.e., are topologically homeomorphic to a sphere). The simple polyhedra of chemical interest generally have the additional property of convexity. All vertices of a convex polyhedron lie entirely on one side of each of its faces so that a convex polyhedron can be laid on a flat tabletop with any face down. Convexity is not a topological property, for convex polyhedra in general can be continuously transformed into nonconvex polyhedra. 

Setting up a Riemannian metric of class C7 on a compact two-dimensional ori-entable manifold transforms this manifold into a Riemann surface. The charts on it are the local charts of the coordinates isothermic for this metric. The Riemannian metric on the sphere generates a Riemann surface which is homeomorphic... 

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