Shape of the universe
From Academic Kids

The shape of the universe is a subject of investigation within cosmology. Cosmologists and astronomers describe the geometry of the observable universe which includes both local geometry and global geometry of the whole universe, which is in practice loosely termed topology, even though strictly speaking it goes beyond topology.
The shape of the universe is not concerned with a geometry near to a dense mass. Rather, the geometries under investigation describe a universe where the average density of mass is evenly distributed. Notwithstanding that the universe is "weakly" inhomogeneous and anisotropic in the largescale structure of the cosmos, both astronomical and cosmological measurements determine the observable universe to be, on average, homogeneous, isotropic and an expanding, or accelerating, universe. Although the geometries within the observable universe are generated by a theory of relativity based on spacetime intervals, local geometry turns out to be describable by the familiar geometries of three spatial dimensions. In turn, the local geometry, together with direct observation and further measurement, is used to constrain the possibilities for the global geometry to be a topology of three spatial dimensions.
One of the investigations within the study of local geometry is:
 to evaluate the curvature of the geometry, currently constrained to be constant and very close to zero  corresponding to a "nearly" flat curvature of space
Two investigations within the study of global geometry are:
 whether the universe is infinite in extent or is a compact space
 whether the universe has a simply or nonsimply connected topology
Contents 
Local geometry
The Local geometry is the geometry describing the observable universe.
Many astronomical observations show the observable universe to be homogeneous and isotropic and infer it to be accelerating.
Both astronomical measurements of supernova events and cosmological measurements of the structures in the ambient cosmic background radiation, are well approximated by a FriedmannLemaîtreRobertsonWalker model. The FLRW model describes a homogeneous, isotropic and accelerating universe in the context of General Relativity. An attraction of the FLRW is that the associated Friedman equations do yield a geometry of the universe as a function of well understood mathematics for fluid dynamics. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe.
Curvature (local geometry)
The homogeneous and isotropic universe allows for a geometry with a constant curvature.
One aspect of local geometry to emerge from General Relativity and the FLRW model is that the critical density is related to the curvature of the geometry of space. Expressing the ratio of average energy density of the universe over critical energy density as Ω, the curvature is given as:
 <math>
curvature = \begin{cases} zero, & \mbox{if the critical density, } \Omega, = 1 \\ positive, & \mbox{if } \Omega > 1 \\ negative, & \mbox{if } \Omega < 1 \end{cases}
<math>
Astronomical measurements of both mass density of the universe and spacetime intervals using supernova events constrain the curvature to be zero or positive but near to zero. Cosmological measurements infer the curvature as close to zero  either zero or just positive or negative.
Feasible local geometries
The many feasible geometries of a constant curvature are categorized into three, according to the sign of the curvature:
 <math> \mbox{local geometry} =
\begin{cases} \mbox{flat}, & \mbox{if the curvature is exactly zero} \\ \mbox{spherical}, & \mbox{if the curvature is positive} \\ \mbox{hyperbolic}, & \mbox{if the curvature is negative} \end{cases}
<math>
An observable universe that is "nearly flat", from the measured curvature, allows for a simplification whereby the dynamic, accelerating dimension of the geometry can be separated and omitted by invoking comoving coordinates. Comoving coordinates, from a single frame of reference, leave a static geometry of three spatial dimensions.
Of eight feasible geometries given by the geometrization conjecture, the shape of the observable universe, or the local geometry, is in all likelihood described by one of the three "primitive" geometries:
 3dimensional Euclidean geometry, generally annotated as E^{3}
 3dimensional spherical geometry with a small curvature, often annotated as S^{3}
 3dimensional hyperbolic geometry with a small curvature, often annotated as H^{3}
Even if the universe is not exactly flat, the curvature is close enough to zero to place the radius beyond the horizon of the observable universe.
Global geometry
Global geometry covers the geometry, in particular the topology, of the whole universe  both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. For a flat local geometry, the scale of any properties of the topology is arbitrary and may or may not be directly detectable. For spherical and hyperbolic geometries, the probability of detection of the topology by direct observation depends on the curvature. Using the radius of the curvature as a scale, a small curvature of the local geometry, with a corresponding scale greater than the observable horizon, makes the topology difficult to detect. A spherical geometry may well have a radius that can be detected. In a hyperbolic geometry the radius scale is unlikely to be within the observable horizon.
Compactness of the global shape
A compact space is a general topological definition that encompasses the more applicable notion of a bounded metric space. If the 3manifold of a spatial section of the universe is compact then, as on a sphere, distances extended far enough "in the same direction" will return to the start point and the space will have a definable "volume" or "scale". If the geometry of the universe is not compact, then it is infinite in extent with infinite paths of constant direction that, generally do not return and the space has no definable volume and an arbitrary scale, such as the Euclidean plane.
If the local geometry is spherical, the topology is compact. Otherwise, a flat or a hyperbolic local geometry, the topology can be either compact or infinite. Even given that the local geometry is nearly flat, the universe may still be bounded  as in a three dimensional version of a cylinder that is also a flat geometry.
One of the endeavors in the analysis of data from the Wilkinson Microwave Anisotropy Probe (WMAP) is to detect multiple "backtoback" images of the distant universe in the cosmic background radiation. Assuming the light has enough time since its origin to travel around a bounded universe, multiple images may be observed. While current results and analysis do not rule out a bounded topology, if the universe is bounded then the curvature is extremely small.
Connectedness of the global manifold
A simply connected space is all of one piece, such as a sphere. A "multiply connected", or more strictly a nonsimply connected space has circleshaped "holes" or "handles". If the global geometry is nonsimply connected then at some points in the geometry, near a the junction of a "handle", paths of light may reach an observer by two routes, a "closed path"  a path through the main body and a path via the handle.
The probability of detecting a multiply connected topology depends not only on the scale of the topology but also the degree of complication in the topology. The observer may have a privileged position near a closed path. In a hyperbolic local geometry, a nonsimply connected space is unlikely to be detected unless the observer is near a closed path.
A second endeavor in the analysis of data from the WMAP is to separate any multiple images in the cosmic background radiation from potential "backtoback" multiple images due to a compact space. Current results eliminate many forms of nonsimply connected topologies implying that either we are not near a closed light path or the global geometry is simply connected.
See also
Theorema Egregium The "remarkable theorem" discovered by Gauss that the inhabitants of a line, a surface or any other differentiable manifold can intrinsically measure the curvature of the space that they inhabit.
Extra dimensions in String Theory for 10 or 11 extra spacelike dimensions all with a compact topology.
External links
 Oct 2003  Poincaré dodecahedral model  WMAP statistics (http://arXiv.org/abs/astroph/0310253) model predictions for a spherical local geometry
 Feb 2004  Poincaré dodecahedral model  11 degree matched circles (http://arXiv.org/abs/astroph/0402608)es:Forma del universo
fr:forme de l'Univers pl:Kształt Wszechświata simple:Shape of the universe sl:Oblika Vesolja