Different Kinds of Metric
Are there different kinds of metric in GR? For instance. I read in http://www.astronomy.ohiostate.edu/...682/notes3.pdf there the FRW Metric is about:
"In 1917, Einstein introduced the first modern cosmological model, based on GR, in which the spatial metric is that of a 3sphere:" Why. Can the GR metric be based on 2D, 3D, 3sphere, 3cube? Can you give other examples? What would happen to the EFE if you change the metric? Btw.. is the FRW metric about space or about spacetime? 
Re: Different Kinds of Metric
There are two ways to classify different metrics:
(1) different solutions (2) different coordinates Think about the equation ax˛+bx+c=0. There are two solutions x_{1,2} (this corresponds to 1) and of course you can make a change of coordinates x'=xa (this corresponds to 2) 
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no;
and what is "the normal GR metric"? 
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And there's the Schwarzschild metric which I think is 4D. I think what's unique about the FRW metric is its 3D or our space compared to others.

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The FRW metric is a 4D metric. The source you cited is only talking about "the spatial part" of the metric.

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Let's take this definition at wiki for a good start:
"In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric. When a topological space has a topology that can be described by a metric, we say that the topological space is metrizable. In differential geometry, the word "metric" is also used to refer to a structure defined only on a differentiable manifold which is more properly termed a metric tensor (or Riemannian or pseudoRiemannian metric)." I'd like to know what is the difference between the FRW Metric in the universe and say the Schwarzschild metric in black hole. Just an intuitive grasp or simple distinctions will do. 
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A particular metric gives you the distance between infinitessimally close points. The FRW and Schwarzschild metrics are two different metrics which are both solutions for the EFE's for two different conditions. Schwarzschild metric is a so called vacuum solution which is valid outside a spherical mass distribution. FRW metric is a solution for a homogeneous and isotropic distribution of dust (perfect fluid).

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The FRW metric is definitely 4D.
[tex]ds^2=c^2dt^2+a(t)^2(\frac{dr^2}{1kr^2}+r^2d\Omega^2)[/tex] Notice the presence of dt in there. 
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How about a black hole, is its spatial metric also that of a 3sphere? What other examples where the spatial metric is not a 3sphere? 
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The metric of spacetime induces a Riemannian metric on each of these hypersurfaces. It's convenient to use these metrics to distinguish between the main classes of FLRW solutions. 
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[tex]dl^2=\left(1\frac{2M}{r}\right)^{1}dr^2+r^2d\Omega^2[/tex] In the case of negative curvature of the universe, the spatial metric is not a 3sphere either. In the case of a flat universe, the spatial metric is not a 3sphere. The spatial part of the Alcubierre metric is not a 3sphere. I could keep going. 
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