Wolfgang, answer to your questions are in the paper "Illusion of space expansion ....." on this site. and other papers by us. Take a look 

Hi J.G. your paper impressively describes the concequences of a Lobachevskian space, which are quite similar to the consequences of the rotational symmetric space model I am proposing. As I understand "Lobachevskian" means hyperbolic and  includes any hyperbolic geometry. (Please correct me if I am wrong)

What I am proposing is a specific finite and isotropic geometry which partially has similar differential geometrical properties as a Lobachewskian space. But the isotropic geometry I am proposing seems to have (as far as I am able to evaluate it) a zero Riemannian curvature. But nevertheless the numerical evaluation leads to laterally diverging geodesics. The impression of a longitudinal dilatation only happens in this geometry because the receiver of a transported volume gets it with a rotated orientation compared with the orientation as it had been filled with radiation. In a Lobachevskian geometry, dilatation in any direction is an intrinsic property.

But I fully agree to the physical consequences of a hyperbolic geometry you are describing in your paper. And I am glad, that your paper substantiates the possibility that transport in a curved geometry can lead to the impression of an expanding space.

Woffgang, Lobachevskian space is a metric , simply connected, non compact space of constant negative Gaussian curvature. There are compact hyperbolic spaces which in the sense of the above definition are not  Loabchevskian. You may take a look at Thurston book, " Three dimensional geometry and topology". That is an excellent book which I would recommend to anybody having interest in geometry.

Hi, J.G., thank you for the reference to the indeed excellent book on three dimensional manifolds. Unfortunately the manifold given by unit quaternions has not been discussed in that book. Do you know a reference which considers transport in a geometry where displacement is mapped to quaternion multiplication?

You can find applications of quaternions  in the book by J.Elstrod, F.Grunewald, J.Mennic\ke,  "Groups Acting On Hyperbolic Space"  Springer

Hi, J.G., thank you again for a surely helpful reference to study general geometric relations. But I suspect that the specific case of a homogenous and isotropic geometry has not been discussed in that book. I further suspect that a Lobachevskian geometry cannot be homogenous and isotropic as required for the geometry of the universe.

Wlifgang, your suspicions are unfounded. Lobachevskian spaces  fall in the category of so called HOMOGENEOUS SPACES  with transitive groups of isometries called also rigid motions.  In Lobachevskian space , neighborhood of any point looks like neighborhood of any other point.  I suggest you study Lobachevskian geometry and then you get a better understanding what it is about. This will require some effort on your part but no doubt will be beneficial for your research.

Hi, J.G, I already studied many geometries but the only geometry I found to be locally Euclidian, finite, homogenous and isotropic is the geometry given by unit quaternions.

I have not been able to identify any  Lobachevskian geometry with this properties. But of course  I would appreciate a specific description of such a geometry.

That sounds strange since quaternionic model of Lobachevskian geometry exists and it is isomorphic to other models. This is clearly shown in the first chapter of the book " Groups Acting on Hyperbolic Space" I recommended to you. Those things  are already known at least 100 years. That is as far as I can help you..  Sorry.

Hi J.G. it seems to be a misunderstanding. My questions are not related  to the local properties of the Lobachevskian geometry which you have comprehensively described in your article. I would like to know the topological properties of the geodesics. Are the geodesics in your model closed in the sense, that if endlessly following an individual geodesic you will finally arrive at your starting point?

Geodesics in Lobachevskian space are defined by two points at +- infinity  .  .

Infinity is just the problem. We need a finite solution. If a Lobachevskian geometry cannot provide closed geodesics it can't be a candidate for a geometry of the universe.

WHY?

There are many reasons to exclude infinity from examinations concerning reality.

On the way to and from inifinity structure does not really matter. Infinity does not have a specific relation to reality. A numerical description of a model cannot include infinity. It will never be possible to validate descriptions/formulas in which infinity occurs even in only one measurable aspect...

Dear Wolfgang Konle,

In cosmology, we use the comoving coordinates and therefore, the  particles are always static. The role of the geodesic is trivial.