This paper : Hyperbolic Universes with a Horned Topology and the CMB Anisotropy http://arxiv.org/astro-ph/0403597 ...press release: http://www.newscientist.com/news/news.jsp?id=ns99994879 proposes a universe with the shape of a horn. This is a hyperbolic space with negative curvature. The paper mentions an interesting issue: the relation between finite hyperbolic spaces and chaos. Finite hyperbolic spaces generate chaotic mixing of trayectories, leading to fractal structure formation. See e.g.: Chaos and order in a finite universe http://arxiv.org/abs/astro-ph/9907288 A fractal nature of large scale structures was already suggested due to the self-similarity of the distribution of galaxies and clusters (similar correlation functions AFAIK). My knowledge of chaotic systems is almost non existent, thus I would like to know qualitatively why finite hyperbolic spaces do have such properties in relation to chaos and infinite flat spaces do not (although you can find an interesting remark in the previous cited paper about the cosmological constant in infinite flat spaces). But there is another thing that bothers me. In the paper it is claimed that the CMB data would not reflect the negative curvature. But why? Usually it is assumed that the angular scale of the first peak of the CMB anisotropies gives a measure of the curvature. Regards.