The Ricci curvature tensor, also simply known as the Ricci tensor (Parker and Christensen 1994), is defined by
where 
is the Riemann tensor.
Geometrically, the Ricci curvature is the mathematical object that controls the growth
rate of the volume of metric balls in a manifold.
See also
Bishop's Inequality,
Campbell's Theorem,
Einstein Tensor,
Milnor's
Theorem,
Riemann Tensor,
Scalar
Curvature
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References
Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation.
San Francisco: W. H. Freeman, 1973.Parker, L. and Christensen,
S. M. "The Ricci, Einstein, and Weyl Tensors." §2.7.1 in MathTensor:
A System for Doing Tensor Analysis by Computer. Reading, MA: Addison-Wesley,
pp. 30-32, 1994.Wald, R. M. General
Relativity. Chicago, IL: University of Chicago Press, p. 40, 1984.Weinberg,
S. Gravitation
and Cosmology: Principles and Applications of the General Theory of Relativity.
New York: Wiley, pp. 135 and 142, 1972.Referenced on Wolfram|Alpha
Ricci Curvature Tensor
Cite this as:
Weisstein, Eric W. "Ricci Curvature Tensor."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RicciCurvatureTensor.html
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