general relativity - Ricci scalars for space and spacetime, local and global curvature

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?

   
   

Answer

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.