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  1. If Ricci scalar describes the full spacetime curvature, then what do we mean by $k=0,+1,-1$ being flat, positive and negative curved space?

  2. Is $k$ special version of a constant "3d-Ricci" scalar?

  3. What is the difference between the local and global spacetime curvature?

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$k$ is a constant appearing in the FLRW metric for a homogenous and isotropic universe, relating to the scalar curvature of spatial slices. $k$ is a special feature of certain cosmological models but the Ricci scalar exists for any spacetime. –  Michael Brown Apr 17 '13 at 14:35

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The $k$ notation is generally used to describe Friedmann Robertson Walker cosmological models. These are built on the assumptions of homogeneity and isotropy. The spacetime can be described as being foliated by spatial slices of constant curvature. The k value is the sign of this spatial curvature if the {-1, 0, +1} convention is adopted. As the curvature is a constant, it makes sense to talk of its sign. Further details here.

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is there any meaningful connection between K and the Ricci tensor? or is it overly complicated to use in the FRW metric? –  Dmist Apr 17 '13 at 14:47
    
If I understood you correctly, the k value is simply the sign of R_ii? That is, for a homogenous and isotropic spacetime, when R_ii is constant. –  Winnie Apr 17 '13 at 14:48
    
@Winnie yes, the sign of $^3R$ –  twistor59 Apr 17 '13 at 14:51
    
@Dmist I guess you mean the full 4 dimensional Ricci tensor? I'm not sure that there is, I can't remember the expressions for the Ricci tensor for FRW off by heart. The full spacetime has an extrinsic curvature which describes how the three-slices are embedded, and this will complicate things. –  twistor59 Apr 17 '13 at 14:55
    
Being the scalar curvature, is k also preserved under the coordinate transformations? (Coordinate transformations of a FRW metric.) –  Winnie Apr 17 '13 at 14:57

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