Interpreting Einstein Tensor Geometrically on a Manifold

ChrisVer
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Do you know how could I interpret the Einstein Tensor geometrically (on a general manifold)?
For example the Christoffel Symbols can show someone the divergence/convergence of geodesics and/or show how the change of metric from point to point creates an additional force/potential (through the equation of geodesics), as well as how we can parallel transpose a vector (since they appear in the covariant derivative as connections)
I guess this should go all the way up to interpreting the Riemann curvature tensor?
 
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