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What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, what is shape of manifold at this point? and a same question about scalar curvature.

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3… ... I suggest making an edit to your post on the other site, so that you can receive better help. – Chris Gerig Apr 4 '13 at 18:56

You find in Wikipedia:

  • "Indeed, if $\xi$ is a vector of unit length on a Riemannian n-manifold, then $Ric(\xi,\xi)$ is precisely (n−1) times the average value of the sectional curvature, taken over all the 2-planes containing $\xi$."

  • In Riemann normal coordinates, the Taylor expansion of the Riemannian volume has vanishing first order term, and the second order term is $1/6$ times the Ricci curvature.

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