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In differential geometry and general relativity space is said to be flat if the Riemann tensor $R=0$. If the Ricci tensor on manifold $M$ is zero, it doesn't mean that the manifold itself is flat. So what's the geometrical meaning of Ricci tensor since it's been defined with the Riemann tensor as

$$\mathrm{Ric}_{ij}=\sum_a R^a_{iaj}?$$

Gold
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Apogee
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2 Answers2

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The local geometric structure of a pseudo-Riemannian manifiold $M$ is completely described by the Riemann tensor $R_{abcd}$. The local structure of a manifold is affected by two possible sources

  1. Matter sources in $M$: The matter distribution on a manifold is described by the stress tensor $T_{ab}$. By Einstein's equations, this can be related to the Ricci tensor (which is the trace of the Riemann tensor = $R_{ab} = R^c{}_{acb}$. $$ R_{ab} = 8 \pi G \left( T_{ab} + \frac{g_{ab} T}{2-d} \right) $$

  2. Gravitational waves on $M$. This is described by the Weyl tensor $C_{abcd}$ which is the trace-free part of the Riemann tensor.

Thus, the local structure of $M$ is completely described by two tensors

  1. $R_{ab}$: This is related to the matter distribution. If one includes a cosmological constant, this tensor comprises the information of both matter and curvature due to the cosmological constant.

  2. $C_{abcd}$: This describes gravitational waves in $M$. A study of Weyl tensor is required when describing quantum gravity theories.

Prahar
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I've always liked the interpretation you get from the Raychaudhuri equation. It shows you that the Ricci tensor tends to cause geodesics to focus together. If you begin with a family of geodesics with tangent vector $u^a$, you can define the expansion $\theta\equiv \nabla_a u^a$ which measures the rate at which geodesics are spreading out or converging together. As you move along a an integral curve of $u^a$, the Raychaudhuri equation tells you how the expansion changes as a function of curve's parameter, $\lambda$: $$ \frac{d}{d\lambda}\theta = -\frac13\theta^2-\sigma_{ab}\sigma^{ab}+\omega_{ab}\omega^{ab}-R_{ab}u^au^b.$$ $\sigma_{ab}$ is called the shear and is related to the tendency of a cross section of the curves to distort toward and ellipsoid, and $\omega_{ab}$ is the vorticity and describes how the curves twist around each other. The Ricci tensor appears in this equation with a minus sign, so that when $R_{ab}u^au^b$ is positive, it tends to decrease the expansion, which describes focusing of the geodesics.

asperanz
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